Chebyshev-type cubature formulas for doubling weights on spheres, balls and simplexes

TL;DR

This paper establishes the existence and properties of Chebyshev-type cubature formulas for doubling weights on spheres, providing bounds on the number of nodes and their distribution, extending recent spherical design results.

Contribution

It proves the existence of Chebyshev-type cubature formulas for doubling weights on spheres with optimal node bounds and separation, extending recent spherical design research.

Findings

Existence of cubature formulas with N nodes for doubling weights.

Optimal bounds on the minimal number of nodes for such formulas.

Construction of regular convex partitions of the sphere with respect to the weight.

Abstract

This paper proves that given a doubling weight $w$ on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$, there exists a positive constant $K_w$ such that for each positive integer $n$ and each integer $N\geq \max_{x\in \mathbb{S}^{d-1}} \frac {K_w} {w(B(x, n^{-1}))}$, there exists a set of $N$ distinct nodes $z_1,\cdots, z_N$ on $\mathbb{S}^{d-1}$ which admits a strict Chebyshev-type cubature formula (CF) of degree $n$ for the measure $w(x) d\sigma_d(x)$, $$ \frac 1{w(\mathbb{S}^{d-1})} \int_{\mathbb{S}^{d-1}} f(x) w(x)\, d\sigma_d(x)=\frac 1N \sum_{j=1}^N f(z_j),\ \ \forall f\in\Pi_n^d, $$ and which, if in addition $w\in L^\infty(\mathbb{S}^{d-1})$, satisfies $$\min_{1\leq i\neq j\leq N}\mathtt{d}(z_i,z_j)\geq c_{w,d} N^{-\frac1{d-1}}$$ for some positive constant $c_{w,d}$. Here, $d\sigma_d$ and $\mathtt{d}(\cdot, \cdot)$ denote the surface Lebesgue measure and the geodesic distance on $\mathbb{S}^{d-1}$ respectively, $B(x,r)$ denotes the spherical cap with center $x\in\mathbb{S}^{d-1}$ and radius $r>0$, $w(E)=\int_E w(x) \, d\sigma_d(x)$ for $E\subset\mathbb{S}^{d-1}$, and $\Pi_n^d$ denotes the space of all spherical polynomials of degree at most $n$ on $\mathbb{S}^{d-1}$. It is also shown that the minimal number of nodes $\mathcal{N}_{n} (wd\sigma_d)$ in a strict Chebyshev-type CF of degree $n$ for a doubling weight $w$ on $\mathbb{S}^{d-1}$ satisfies $$\mathcal{N}_n (wd\sigma_d) \sim \max_{x\in \mathbb{S}^{d-1}} \frac 1 {w(B(x, n^{-1}))},\ \ n=1,2,\cdots.$$ Proofs of these results rely on new convex partitions of $\mathbb{S}^{d-1}$ that are regular with respect to a given weight $w$ and integer $N$. Our results extend the recent results of Bondarenko, Radchenko, and Viazovska on spherical designs ({\it Ann. of Math. (2)} {\bf 178}(2013), no. 2, 443--452,{\it Constr. Approx.} {\bf 41}(2015), no. 1, 93--112).

Equations

$$\underset{1\le i\ne j\le N}{\min }\arccos (z_i\cdot z_j)\ge c_{w,d}{N}^{-\frac{1}{d-1}}>0.$$

$${\mathcal{N}}_n(wd{\sigma }_d)\sim \underset{x\in {\mathbb{S}}^{d-1}}{\max }\frac{1}{w(B(x,n^{-1}))},n=1,2,\cdots .$$

$$\frac{1}{\mu (\Omega )}{\int }_{\Omega }P(x)d\mu (x)=\frac{1}{N}\sum _{j=1}^{N}P(z_j)$$

$${\mathcal{N}}_n(d{\mu }_{\alpha ,\beta })\sim n^{2+2\max \{\alpha ,\beta \}},n=1,2,\cdots .$$

$$\Pi_{n}^{d}:=\text{span}\Bigl{\{}x^{\alpha}=x_{1}^{{\alpha}_{1}}\cdots x_{d}^{{\alpha}_{d}}:\ \ {\alpha}=({\alpha}_{1},\cdots,{\alpha}_{d})\in{\mathbb{N}}_{0}^{d},\ \ |{\alpha}|:=\sum_{j=1}^{d}{\alpha}_{j}\leq n\Bigr{\}},$$

$$w(B(x,2r))\le Lw(B(x,r))\forall x\in {\mathbb{S}}^{d-1},\forall r>0,$$

$$w(B(x,\lambda t))\le 2{\lambda }^{s_w}w(B(x,t)),\forall \lambda >1,\forall t>0,\forall x\in {\mathbb{S}}^{d-1}.$$

$$w_{\alpha }(x)=\prod _{j=1}^d|x_j{|}^{{\alpha }_j},x\in {\mathbb{S}}^{d-1},\alpha =({\alpha }_1,\cdots ,{\alpha }_d)\in (-1,\infty {)}^d.$$

$${\int }_{{\mathbb{S}}^{d-1}}P(x)w(x)d{\sigma }_d(x)=\frac{1}{N}\sum _{j=1}^{N}P(z_j),\forall P\in {\Pi }_n^d.$$

$$\underset{1\le i\ne j\le N}{\min }\mathtt{d}(z_i,z_j)\ge c_{\ast }{N}^{-\frac{1}{d-1}},$$

$${\mathcal{N}}_n(wd{\sigma }_d)\sim \underset{x\in {\mathbb{S}}^{d-1}}{\max }\frac{w({\mathbb{S}}^{d-1})}{w(B(x,n^{-1}))},n=1,2,\cdots$$

$$w_{\alpha }(B(x,\theta ))\sim {\theta }^{d-1}\prod _{j=1}^d(|x_j|+\theta {)}^{{\alpha }_j},x\in {\mathbb{S}}^{d-1},\theta \in (0,\pi ].$$

$${\mathcal{N}}_n(w_{\alpha }d{\sigma }_d)\sim n^{d-1+{\sum }_{j\in I_{\alpha }}{\alpha }_j-\max \{{\alpha }_{\min },0\}},$$

$$x{\tau }_j=(x_1,\cdots ,x_{j-1},-x_j,x_{j+1},\cdots ,x_d),x\in {\mathbb{R}}^d.$$

$$\rho_{{\Omega}}(x,y):=\begin{cases}\|x-y\|+\bigl{|}\sqrt{x_{d+1}}-\sqrt{y_{d+1}}\bigr{|},&\ \ \text{if ${\Omega}={\mathbb{B}}^{d}$},\\ \arccos(\sum_{j=1}^{d+1}\sqrt{x_{j}y_{j}}),&\ \ \text{if ${\Omega}={\mathbb{T}}^{d}$},\end{cases}$$

$$x_{d+1}=\left\{\begin{array}[]{ll}\sqrt{1-\|x\|^{2}},&\hbox{$x\in{\mathbb{B}}^{d}$;}\\ 1-|x|=1-x_{1}-\cdots-x_{d},&\hbox{$x\in{\mathbb{T}}^{d}$.}\end{array}\right.$$

$$w(B_{\Omega }(x,2r))\le Lw(B_{\Omega }(x,r)),\forall x\in \Omega ,\forall r>0$$

$${\varphi }_{\Omega }(x)=\{\begin{array}{ll}(x_1,x_2,\cdots ,x_d),& \text{if Ω=Bd},\\ (x_1^2,x_2^2,\cdots ,x_d^2),& \text{if Ω=Td},\end{array}x=(x_1,x_2,\cdots ,x_d,x_{d+1})\in {\text{SS}}^d.$$

$${\int }_{\Omega }f(x)dx=c_d{\int }_{{\text{SS}}^d}f({\varphi }_{\Omega }(x)){\Phi }_{\Omega }(x)d{\sigma }_{d+1}(x),$$

$$w_{\Omega }(x,x_{d+1})=\{\begin{array}{ll}w(x)|x_{d+1}|,& \text{if Ω=Bd},\\ ({\prod }_{j=1}^{d+1}|x_j|)w(x_1^2,\cdots ,x_d^2),& \text{if Ω=Td},\end{array}$$

$$E^{\Omega }=\{\begin{array}{ll}\{(x,\sqrt{1-\Vert x{\Vert }^2}):x\in E\},& \text{ if Ω=Bd},\\ \{(\sqrt{x_1},\cdots ,\sqrt{x_d},\sqrt{1-|x|}):x\in E\},& \text{ if Ω=Td}.\end{array}$$

$${\int }_{\Omega }P(x)w(x)dx=\frac{1}{N}\sum _{j=1}^{N}P(z_j),\forall P\in {\Pi }_n^d.$$

$$S=\Bigl{\{}g(x)=\frac{x}{\|x\|}:\ \ x=\sum_{j=1}^{d}t_{j}\xi_{j},\ \ t_{1},\cdots,t_{d}\geq 0\ \ \text{and }\ \ \sum_{j=1}^{d}t_{j}=1\Bigr{\}}.$$

$${\int }_{S}F(x)d{\sigma }_d(x)={\int }_{T}F(g(y))\frac{a}{\Vert y{\Vert }^d}dy,$$

$$\frac{a}{2(a+1)}\Vert x-y\Vert \le \mathtt{d}(g(x),g(y))\le \frac{\pi }{2a}\Vert x-y\Vert ,\forall x,y\in H_T\cap 2{\mathbb{B}}^d.$$

$${\sin }^2\frac{\mathtt{d}(x,y)}{2}={\sin }^2\frac{{\theta }_1-{\theta }_2}{2}+(\sin {\theta }_1\sin {\theta }_2){\sin }^2\frac{\mathtt{d}({\eta }_1,{\eta }_2)}{2}.$$

$$|{\theta }_1-{\theta }_2|$$

$$\mathtt{d}(x,y)$$

$$c{r}^{d-1}a\le m_d(G)={\int }_0^{2r}{m}_{d-1}(G(t))dt.$$

$$R_j:=\{(\cos \theta ,\xi \sin \theta ):\xi \in E_j,0\le \theta \le \pi \},j=1,2,\cdots ,M.$$

Full text

[TABLE]

where $d\sigma_{d}$, $B(x,r)$ and $\Pi_{n}^{d}$ denote the surface Lebesgue measure on ${\mathbb{S}^{d-1}}$, the spherical cap with center $x\in\mathbb{S}^{d-1}$ and radius $r>0$, and the space of all spherical polynomials of degree at most $n$ on ${\mathbb{S}^{d-1}}$, respectively, and $w(E)=\int_{E}w(x)\,d\sigma_{d}(x)$ for $E\subset{\mathbb{S}^{d-1}}$. If, in addition, $w\in L^{\infty}({\mathbb{S}^{d-1}})$, then the above set of nodes can be chosen to be well separated:

[TABLE]

It is further proved that the minimal number of nodes $\mathcal{N}_{n}(wd{\sigma}_{d})$ required in ($\ast$) for a doubling weight $w$ on ${\mathbb{S}^{d-1}}$ satisfies

[TABLE]

Proofs of these results rely on new convex partitions of ${\mathbb{S}^{d-1}}$ that are regular with respect to a given weight $w$ and integer $N$. Similar results are also established on the unit ball and the standard simplex of $\mathbb{R}^{d}$.

Our results extend the recent results of Bondarenko, Radchenko, and Viazovska on spherical designs.

This work was supported by NSERC Canada under grant RGPIN 04702. It was conducted when the second author was a Ph.D student at the University of Alberta.

1. Introduction

A Chebyshev-type cubature formula (CF) of degree $n$ for a positive Borel measure $\mu$ on a compact subset ${\Omega}$ of ${\mathbb{R}}^{d}$ consists of finitely many nodes $z_{1},\cdots,z_{N}$ in ${\Omega}$ (which may repeat) such that the numerical integration formula

[TABLE]

holds for every algebraic polynomial $P$ of total degree at most $n$ in $d$ variables, where $N$ is called the size of the formula. A Chebyshev-type CF is called strict if all the nodes $z_{j}$ are distinct. We denote by $\mathcal{WN}_{n,{\Omega}}(d\mu)$ (resp., $\mathcal{N}_{n,{\Omega}}(d\mu)$) the minimal size of the Chebyshev-type CF (resp., the strict Chebyshev-type CF) of degree $n$ for the measure $\mu$ on ${\Omega}$. We will drop the subscript ${\Omega}$ here whenever the underlying domain ${\Omega}$ is easily understood from the context and no confusion is possible.

A central question concerning the Chebyshev-type CF is to find sharp asymptotic estimates of the quantity $\mathcal{N}_{n,{\Omega}}(d\mu)$ or $\mathcal{WN}_{n,{\Omega}}(d\mu)$ as $n\to\infty$. Study of this question starts with the classical work of Bernstein [2, 3] who shows that $\mathcal{WN}_{n}(dx)\sim n^{2}$ for the Lebesgue measure $dx$ on the interval $[-1,1]$. Here and throughout the paper, the notation $a_{n}\sim b_{n}$ means that $c_{1}a_{n}\leq b_{n}\leq c_{2}a_{n}$ for some positive constants $c_{1},c_{2}$ independent of $n$ (called constants of equivalence).

Bernstein’s methods have been extended and developed in a series of papers of Kuijlaars (see, for instance, [20, 21, 22, 23, 24]), who, in particular, proves that for the Jacobi measures $d\mu_{{\alpha},{\beta}}(t)=(1-t)^{\alpha}(1+t)^{\beta}\,dt$ on $[-1,1]$ with nonnegative parameters ${\alpha},{\beta}\geq 0$,

[TABLE]

Kuijlaars [20, 21] also noticed that his techinique in general does not work for the Jacobi measures $d\mu_{{\alpha},{\beta}}(t)$ with negative parameters ${\alpha},{\beta}>-1$, although he was able to prove a stronger result in [25] implying the estimate (1.2) for the case of ${\alpha}=-\frac{1}{2}$, $-\frac{1}{2}-\lambda_{0}\leq{\beta}<-\frac{1}{2}$ and some positive constant $\lambda_{0}$. The estimate (1.2) was proved recently by Kane[17] for ${\alpha},{\beta}\geq-\frac{1}{2}$, and by Gilboa and Peled [14] for the general case of ${\alpha},{\beta}>-1$. The very interesting work of Gilboa and Peled [14] also uses the method of Kane[17] to establish sharp bounds on the size of the Chebyshev-type CFs in one dimension for all doubling weights with some excellent discussions on non-doubling case. For more information on the Chebyshev-type CFs in one variable, we refer to [15, 16, 13, 18, 32] and references within.

For the Chebyshev-type CFs in several variables, the most well studied case is that of spherical designs, introduced by Delsarte, Goethals, and Seidel [1] in 1977. Let $\mathbb{S}^{d-1}:=\{x\in{\mathbb{R}}^{d}:\ \ \|x\|=1\}$ denote the unit sphere of ${\mathbb{R}}^{d}$ equipped with the usual surface Lebesgue measure $d{\sigma}_{d}(x)$, where $\|\cdot\|$ denotes the Euclidean norm. A spherical $n$-design on $\mathbb{S}^{d-1}$ is a Chebyshev-type CF of degree $n$ for the measure $d{\sigma}_{d}(x)$ on $\mathbb{S}^{d-1}$. It is not difficult to show a spherical $n$-design on $\mathbb{S}^{d-1}$ must have size at least $c_{d}n^{d-1}$; that is, $\mathcal{N}_{n}(d{\sigma}_{d})\geq c_{d}n^{d-1}$, (see, for instance, [1]). It was conjectured by Korevaar and Meyers [19] that $\mathcal{N}_{n}(d{\sigma}_{d})\leq c_{d}n^{d-1}$. Much work had been done towards this conjecture (see [17] and the references thererin). This conjecture was recently confirmed in the breakthrough work of Bondarenko, Radchenko, and Viazovska [5, 6], which shows that there exist positive constants $K_{d}$ and $c_{d}$ depending only on $d$ such that given every integer $N>K_{d}n^{d-1}$ there exists a spherical $n$-design consisting of $N$ distinct nodes $z_{1},\cdots,z_{N}\in\mathbb{S}^{d-1}$ with $\min_{1\leq i\neq j\leq N}\mathtt{d}(z_{i},z_{j})\geq c_{d}N^{-\frac{1}{d-1}}$.

In the above mentioned interesting paper [17], Kane develops techniques different from those of [5, 6] to establish bounds on the size of Chebyshev-type CFs on rather general path-connected topological spaces. In particular, his techniques can be applied to prove the existence of spherical $n$-designs on $\mathbb{S}^{d-1}$ of size $O_{d}(n^{d-1}(\log n)^{d-2})$, which is only slightly worse than the asymptotically optimal estimate $O_{d}(n^{d-1})$. In the very general setting of path-connected topological spaces, the work of Kane [17] proves the existence of the Chebyshev-type CFs whose size is roughly the square of the optimal size conjectured in [17]. Using the method of [5], Etayo, Marzo and Ortega-Cerdà [12] establish asymptotically optimal bounds on the size of the Chebyshev-type CFs on compact algebraic manifolds, confirming Kane’s conjecture in certain sense for this specific setting.

Some earlier related works regarding the Chebyshev-type CFs on certain multi-variate domains can be found in the papers of Kuperberg [27, 28, 29]. One can also find some interesting results on Chebyshev-type CFs on discrete spaces such as combinatorial designs and Hadamard matrices, in [30] and the references therein.

One of the main purposes in this paper is to extend the methods of Bondarenko, Radchenko, and Viazovska [5, 6] to determine the asymptotically optimal bounds on the size of the Chebyshev-type CFs for doubling weights on the unit sphere $\mathbb{S}^{d-1}$ and other related domains. Our results are mainly for the case of more variables, whereas the results in one variable were mostly established in the recent paper [14].

Let us start with some necessary notation. Denote by $\Pi_{n}^{d}$ the space of all real algebraic polynomials of total degree at most $n$ in $d$ variables:

[TABLE]

where ${\mathbb{N}}_{0}$ denotes the set of all nonnegative integers. Let $\mathtt{d}(x,y)$ denote the geodesic distance on $\mathbb{S}^{d-1}$; that is, $\mathtt{d}(x,y):=\arccos(x\cdot y)$ for $x,y\in\mathbb{S}^{d-1}$. Denote by $B(x,r)$ the spherical cap $\{y\in\mathbb{S}^{d-1}:\ \ \mathtt{d}(x,y)\leq r\}$ with center $x\in\mathbb{S}^{d-1}$ and radius $r>0$. A weight function $w$ on $\mathbb{S}^{d-1}$ (i.e., a nonnegative integrable function on $\mathbb{S}^{d-1}$ ) is said to satisfy the doubling condition if there exists a positive constant $L$ such that

[TABLE]

where we write $w(E):=\int_{E}w(x)\,d{\sigma}_{d}(x)$ for $E\subset\mathbb{S}^{d-1}$, and the least constant $L=L_{w}$ is called the doubling constant of $w$. Given a doubling weight $w$ on $\mathbb{S}^{d-1}$, there exists a constant $d-1\leq s_{w}\leq\frac{\log L_{w}}{\log 2}$ such that

[TABLE]

Many of the weights that appear in analysis on $\mathbb{S}^{d-1}$ satisfy the doubling condition; in particular, all weights of the form

[TABLE]

In this paper, we will prove the following weighted extension of the result of Bondarenko, Radchenko, and Viazovska [5, 6] on spherical designs:

Theorem 1.1**.**

Let $w$ be a doubling weight on $\mathbb{S}^{d-1}$ normalized by $w(\mathbb{S}^{d-1})=1$. Then there exists a positive constant $K_{w}$ depending only on the doubling constant of $w$ and the dimension $d$ such that for each given positive integer $n$, and every integer $N\geq\max_{x\in\mathbb{S}^{d-1}}\frac{K_{w}}{w(B(x,n^{-1}))}$, there exist $N$ distinct nodes $z_{1},\cdots,z_{N}\in\mathbb{S}^{d-1}$ for which

[TABLE]

If, in addition, $w\in L^{\infty}(\mathbb{S}^{d-1})$, then the set of nodes $\{z_{1},\cdots,z_{N}\}$ above can be chosen to be well separated:

[TABLE]

where $c_{\ast}$ is a positive constant depending only on $\|w\|_{\infty}$ and the doubling constant of $w$.

As a corollary of Theorem 1.1, we have

Corollary 1.2**.**

Given a doubling weight $w$ on $\mathbb{S}^{d-1}$, the minimal sizes of the strict Chebyshev-type CFs for the measure $w(x)d{\sigma}_{d}(x)$ on $\mathbb{S}^{d-1}$ satisfy

[TABLE]

with the constants of equivalence depending only on $d$ and the doubling constant of $w$.

Several remarks are in order.

Remark 1.3*.*

1. (i)

By (1.3), it is easily seen that $\max_{x\in\mathbb{S}^{d-1}}\frac{1}{w(B(x,n^{-1}))}\leq 2\pi^{s_{w}}n^{s_{w}}$. Thus, Corollary 1.2 particularly implies that $\mathcal{N}_{n}(wd{\sigma}_{d})\leq Cn^{s_{w}}<\infty$ for every doubling weight $w$ on $\mathbb{S}^{d-1}$. 2. (ii)

For the weights $w_{\alpha}$ given in (1.4), a straightforward calculation shows that

[TABLE]

By Corollary 1.2, this implies that

[TABLE]

where ${\alpha}_{\min}=\min_{1\leq i\leq d}{\alpha}_{i}$ and $I_{{\alpha}}=\{i:\ \ 1\leq i\leq d,\ \ {\alpha}_{i}\geq 0\}$. 3. (iii)

Note that only Chebyshev-type CFs with distinct nodes are involved in Theorem 1.1. However, it is worthwhile to point out that a slight modification of our proof shows that for each integer $N\geq\max_{x\in\mathbb{S}^{d-1}}\frac{K_{w}}{w(B(x,n^{-1}))}$ there exists a Chebyshev-type CF of degree $n$ and size $N$ for the measure $w(x)d{\sigma}_{d}(x)$ consisting of a large number of multiple nodes (i.e., repeated notes), and with the number $N_{n}$ of distinct nodes satisfying $N_{n}\sim n^{d-1}$.

While the proof of Theorem 1.1 follows the methods of the papers [5, 6], it is more technical and involved than the corresponding unweighted case due to the fact that the measure $w(x)d{\sigma}_{d}(x)$ is not rotation-invariant, which means that in general, the weighted measure $w(B(x,r))$ of a spherical cap not only depends on radius $r$ but also on the center $x$.

An important ingredient used in our proof is the convex partition of $\mathbb{S}^{d-1}$ that is regular with respect to a given weight. Recall that a subset $A\subset\mathbb{S}^{d-1}$ is geodesically convex if any two points $x,y\in A$ can be joined by a geodesic arc that lies entirely in $A$, whereas a finite collection $\{R_{1},R_{2},\cdots,R_{N}\}$ of closed geodesically convex subsets of $\mathbb{S}^{d-1}$ is called a convex partition of $\mathbb{S}^{d-1}$ if $\mathbb{S}^{d-1}=\bigcup_{j=1}^{N}R_{j}$ and the interiors of the sets $R_{j}$ are pairwise disjoint. Our result on regular convex partitions of the weighted sphere can be stated as follows:

Theorem 1.4**.**

Let $w$ be a normalized weight on $\mathbb{S}^{d-1}$ (i.e., $w(\mathbb{S}^{d-1})=1$) satisfying that $w(B)>0$ for every nonempty spherical cap $B\subset\mathbb{S}^{d-1}$. Assume that $r\in(0,\pi]$ and $N$ is a positive integer satisfying $\min_{x\in\mathbb{S}^{d-1}}w(B(x,r))\geq\frac{1}{N}$. Then there exists a convex partition $\{R_{1},\cdots,R_{M}\}$ of $\mathbb{S}^{d-1}$ satisfying that $B(x_{j},c_{d}^{\prime}r)\subset R_{j}\subset B(x_{j},c_{d}r)$ and $Nw(R_{j})\in{\mathbb{N}}$ for every $1\leq j\leq M$, where $c_{d},c_{d}^{\prime}$ are two positive constants depending only on $d$.

Several remarks on Theorem 1.4 are in order:

Remark 1.5*.*

1. (i)

Theorem 1.4 seems to be of independent interest. It may have other applications in discrepancy theory and optimization of discrete energies on the sphere (see, for instance, [26, 34]). 2. (ii)

In the case of $w\equiv 1$, Theorem 1.4 with $M=N$ and ${\sigma}_{d}(R_{j})=\frac{1}{N}$ for $j=1,\cdots,N$ is due to Bondarenko, Radchenko, and Viazovska [6, Proposition 1]. The proof of Theorem 1.4 is, however, much more involved than the corresponding case of the surface Lebesgue measure $d{\sigma}_{d}$ due to the fact that the measure $w(x)d{\sigma}_{d}(x)$ is not rotation-invariant. 3. (iii)

Since the weight $w$ may have zeros or discontinuities on $\mathbb{S}^{d-1}$, the integers $k_{j}:=Nw(R_{j})$ in Theorem 1.4 in general depend on the location of $R_{j}$, which is different from the corresponding unweighted case. This difference also increases the technical difficulties of the proof of Theorem 1.1. 4. (iv)

If, in addition, the weight $w$ satisfies the doubling condition, then according to (1.3), there exists a constant $c_{w}$ such that the condition, $\min_{x\in\mathbb{S}^{d-1}}w(B(x,r))\geq\frac{1}{N}$, is satisfied whenever $r\geq c_{w}N^{-1/s_{w}}$.

If we drop the convexity requirement in the partition in Theorem 1.4, we may deduce the following corollary:

Corollary 1.6**.**

Under the conditions of Theorem 1.4, there exists a partition $\{R_{1},\cdots,R_{N}\}$ of $\mathbb{S}^{d-1}$ such that $w(R_{j})=\frac{1}{N}$ and $\text{diam}(R_{j})\leq C_{w}r$ for $j=1,\cdots,N$.

Corollary 1.6 follows directly from Theorem 1.4 due to the fact that if $E\subset\mathbb{S}^{d-1}$ and $0<{\alpha}<w(E)$, then there exists a subset $F$ of $E$ such that $w(F)={\alpha}$.

In order to establish similar results on other domains, we also need to consider weights on $\mathbb{S}^{d-1}$ that are symmetric under certain reflection groups, in which case it can be shown that the set of nodes in the corresponding Chebyshev cubature formula enjoys the same symmetry. Let us first describe briefly some necessary notation. Given $j=1,2,\cdots,d$, we denote by $\tau_{j}$ the reflection with respect to the coordinate plane $x_{j}=0$; that is,

[TABLE]

Denote by ${\mathbb{Z}}_{2}^{d}$ the abelian reflection group generated by the reflections $\tau_{1},\cdots,\tau_{d}$. A weight $w$ on $\mathbb{S}^{d-1}$ is called $\tau_{j}$-invariant for a given $j$ if $w(x\tau_{j})=w(x)$ for all $x\in\mathbb{S}^{d-1}$, and is called ${\mathbb{Z}}_{2}^{d}$-invariant if it is $\tau_{j}$-invariant for every $j=1,2,\cdots,d$. Similarly, we say a finite subset $\Lambda$ of $\mathbb{S}^{d-1}$ is $\tau_{j}$-invariant for a given $j\in\{1,2,\cdots,d\}$ if $\Lambda=\{x\tau_{j}:\ \ x\in\Lambda\}$, whereas it is ${\mathbb{Z}}_{2}^{d}$-invariant if it is $\tau_{j}$-invariant for every $j=1,2,\cdots,d$. For simplicity, we set $\SS_{\text{inter}}^{d-1}:=\{x\in\mathbb{S}^{d-1}:\ \ x_{i}\neq 0,\ \ i=1,2,\cdots,d\}$.

A slight modification of the proof of Theorem 1.1 yields the following result.

Corollary 1.7**.**

Let $w$ be a ${\mathbb{Z}}_{2}^{d}$-invariant (resp. $\tau_{d}$-invariant) doubling weight on $\mathbb{S}^{d-1}$ normalized by $w(\mathbb{S}^{d-1})=1$. Then the conclusions of Theorem  1.1 hold with the set of nodes $\Lambda:=\{z_{1},\cdots,z_{N}\}$ being ${\mathbb{Z}}_{2}^{d}$-invariant (resp. $\tau_{d}$-invariant) and contained in the set $\SS_{\text{inter}}^{d-1}$, (i.e. none of the nodes lies in the coordinate planes).

The weighted results on the sphere also allow us to establish similar results on the unit ball ${\mathbb{B}}^{d}:=\{x\in{\mathbb{R}}^{d}:\ \ \|x\|\leq 1\}$ and on the simplex {\mathbb{T}}^{d}:=\Bigl{\{}x=(x_{1},\cdots,x_{d})\in{\mathbb{R}}^{d}:\ \ x_{1},\cdots,x_{d}\geq 0,|x|:=\sum_{j=1}^{d}|x_{j}|\leq 1\Bigr{\}}. Indeed, it was observed by Y. Xu [36] that CFs on ${\mathbb{B}}^{d}$ and ${\mathbb{T}}^{d}$ are closely related to weighted CFs on $\SS^{d}$. To describe this connection, we use the notation ${\Omega}$ to denote either the unit ball ${\mathbb{B}}^{d}$ or the simplex ${\mathbb{T}}^{d}$ equipped with the usual Lebesgue measure $dx$. Consider the following metric $\rho_{{\Omega}}$ on the domain ${\Omega}$: for $x,y\in{\Omega}$,

[TABLE]

where we write

[TABLE]

Denote by $B_{\Omega}(x,r)$ the ball $\{y\in{\Omega}:\ \ \rho_{\Omega}(x,y)\leq r\}$ with center $x\in{\Omega}$ and radius $r>0$. A doubling weight on ${\Omega}$ is a weight $w$ on ${\Omega}$ satisfying that

[TABLE]

for some positive constant $L$, where $w(E)=\int_{E}w(x)\,dx$ for $E\subset{\Omega}$.

Next, we define the mapping $\phi_{\Omega}:\ \ \SS^{d}\to{\Omega}$ by

[TABLE]

A change of variables shows that for each integrable function $f$ on ${\Omega}$ (see [36])

[TABLE]

where $\Phi_{{\mathbb{B}}^{d}}(x)=|x_{d+1}|$ and $\Phi_{{\mathbb{T}}^{d}}(x)=\prod_{j=1}^{d+1}|x_{j}|$ for $x=(x_{1},\cdots,x_{d+1})\in\SS^{d}$. Note that if $f\in\Pi_{n}^{d}$, then $f\circ\phi_{{\Omega}}$ is a $\tau_{d+1}$-invariant or ${\mathbb{Z}}_{2}^{d}$ invariant spherical polynomial on $\SS^{d+1}$ of degree at most $n$ or $2n$ depending on whether ${\Omega}={\mathbb{B}}^{d}$ or ${\Omega}={\mathbb{T}}^{d}$.

Given a weight $w$ on ${\Omega}$, we define a weight $w_{\Omega}$ on the sphere $\SS^{d}$ by

[TABLE]

where $x\in{\mathbb{B}}^{d}$ and $(x,x_{d+1})\in\SS^{d}$. Clearly $w_{{\Omega}}$ is a $\tau_{d+1}$ invariant or ${\mathbb{Z}}_{2}^{d}$ invariant weight on $\SS^{d}$ satisfying $w_{{\Omega}}(z)=w(\phi_{\Omega}(z))\Phi_{\Omega}(z)$ for $z\in\SS^{d}$. Thus, according to (1.7), $w$ is a doubling weight on ${\Omega}$ if and only if $w_{\Omega}$ is a doubling weight on the sphere $\SS^{d}$, and furthermore, $w(E)=w_{\Omega}(E^{{\Omega}})$ for every measurable $E\subset{\Omega}$, where

[TABLE]

The following are some examples of doubling weights on the ball ${\mathbb{B}}^{d}$ or the simplex ${\mathbb{T}}^{d}$:

Example 1.8**.**

1. (i)

$w(x):=\prod_{j=1}^{m}\|x-\eta_{j}\|^{{\alpha}_{j}}$, $x\in{\mathbb{B}}^{d}$, where ${\alpha}_{1},\cdots,{\alpha}_{m}>-1$ and $\eta_{1},\cdots,\eta_{m}$ are distinct points in ${\mathbb{B}}^{d}$. 2. (ii)

$w(x)=(1-\|x\|)^{{\alpha}_{d+1}}\prod_{j=1}^{d}|x_{j}|^{{\alpha}_{j}}$, $x\in{\mathbb{B}}^{d}$, where ${\alpha}_{1},\cdots,{\alpha}_{d+1}>-1$. 3. (iii)

$w(x)=(1-|x|)^{\mu}\prod_{j=1}^{d}|x_{j}|^{{\kappa}_{j}}$, $x\in{\mathbb{T}}^{d}$, where ${\kappa}_{1},\cdots,{\kappa}_{d},\mu>-1$.

As a direct consequence of (1.7) and Corollary 1.7, we deduce

Corollary 1.9**.**

Let ${\Omega}$ denote either the unit ball ${\mathbb{B}}^{d}$ or the simplex ${\mathbb{T}}^{d}$, and $w$ a doubling weight on ${\Omega}$ normalized by $w({\Omega})=1$. Let $M^{\Omega}_{n,w}:=\max_{x\in{\Omega}}\frac{1}{w(B_{{\Omega}}(x,n^{-1}))}$ for $n=1,2,\cdots$. Then there exist positive constants $K_{d,w}$, $C_{d,w}$, $c_{d,w}$ depending only on $d$ and the doubling constant of $w$ such that the following statements holds:

1. (i)

For $n\in{\mathbb{N}}$, we have $c_{d,w}M^{\Omega}_{n,w}\leq\mathcal{N}_{n,{\Omega}}(w(x)dx)\leq C_{d,w}M^{\Omega}_{n,w}$. 2. (ii)

For each integer $N\geq K_{d,w}M_{n,w}^{{\Omega}}$, there exists a set of $N$ distinct nodes $z_{1},\cdots,z_{N}$ in the interior of ${\Omega}$ such that

[TABLE]

If, in addition, $w_{{\Omega}}\in L^{\infty}(\SS^{d})$, then the set of nodes $z_{1},\cdots,z_{N}\in{\Omega}$ can be chosen so that $\min_{1\leq i\neq j\leq N}\rho_{{\Omega}}(z_{i},z_{j})\geq c_{*}N^{-\frac{1}{d}}$, where $c_{\ast}$ is a positive constant depending only on $\|w_{\Omega}\|_{\infty}$ and the doubling constant of $w$.

The rest of the paper is organized as follows. The first two sections are devoted to the proof of Theorem   1.4. To be more precise, in Section  2 we prove Theorem   1.4 under the additional condition $\frac{N}{2^{d-2}}\in{\mathbb{N}}$, which is technically easier, but already contains some crucial ideas. The proof of Theorem 1.4 for the general case of positive integer $N$ is more complicated and involved, and is given in Section  3. A crucial ingredient used in the proof in Section  3 is the family of nonlinear dilations $T_{\alpha}$ on the sphere that preserve geodesic simplexes. Section  4 contains some preliminary lemmas that are either known or relatively easy to prove, but will be needed in the proof of Theorem 1.1. The final section, Section 5, is devoted to the proofs of Theorem 1.1 and Corollary 1.2. One of the main difficulties in our proofs comes from the fact that the positive integers $Nw(R_{j})$ in Theorem 1.4 ( i.e., the theorem on convex partition) may not be equal to one, which is different from the unweighted case.

For the rest of the paper, we use the notation $C_{w}$, $L_{w}$ , etc. ($c_{w}$ , $\lambda_{w}$, etc.) for sufficiently large (small) constants depending only on the dimension $d$ and the doubling constant of $w$. Sor simplicity, we will always assume that $d\geq 3$ (i.e., $\mathbb{S}^{d-1}\neq\SS^{1}$). The case of $d=2$ (i.e., $\mathbb{S}^{d-1}=\SS^{1}$) can be treated similarly and is, in fact, much simpler.

2. Proof of Theorem 1.4: Case 1.

This section is devoted to the proof of Theorem 1.4 under the additional assumption $\frac{N}{2^{d-2}}\in{\mathbb{N}}$.

2.1. Preliminaries

We start with the concept of geodesic simplex, which will play a crucial role in our proof.

Definition 2.1**.**

A subset $S$ of $\mathbb{S}^{d-1}$ is called a geodesic simplex of $\mathbb{S}^{d-1}$ spanned by a set of linearly independent vectors $\xi_{1},\cdots,\xi_{d}\in\mathbb{S}^{d-1}$ and is denoted by $S=\text{conv}_{\mathbb{S}^{d-1}}\{\xi_{1},\cdots,\xi_{d}\}$ if

[TABLE]

A geodesic simplex $S$ is called admissible if, in addition, the vectors $\xi_{2},\cdots,\xi_{d}$ are mutually orthogonal, $\arccos\xi_{1}\cdot\xi_{2}\in[\frac{\pi}{6},\frac{\pi}{2}]$ and $\xi_{1}\cdot\xi_{j}=0$ for $j=3,\cdots,d$, in which case $\{\xi_{1},\cdots,\xi_{d}\}$ is called an admissible subset of $\mathbb{S}^{d-1}$.

Next, we introduce some necessary notation. For any two distinct points $\xi,\eta\in\mathbb{S}^{d-1}$, we denote by $\operatorname{Arc}(\xi,\eta)$ the geodesic arc connecting $\xi$ and $\eta$; that is, $\operatorname{Arc}(\xi,\eta)=\{t_{1}\xi+t_{2}\eta:\ \ t_{1},t_{2}\geq 0,\ \ t_{1}\xi+t_{2}\eta\in\mathbb{S}^{d-1}\}$. For $x\neq y\in{\mathbb{R}}^{d}$, let $[x,y]$ denote the line segment $\{(1-t)x+ty:\ \ t\in[0,1]\}$. A partition $\{E_{1},\cdots,E_{n}\}$ of a set $E\subset{\mathbb{R}}^{d}$ is called convex if each $E_{j}$ is a convex subset of ${\mathbb{R}}^{d}$. Denote by $\text{conv}_{{\mathbb{R}}^{d}}(E)$ the regular convex hull of a set $E$ in ${\mathbb{R}}^{d}$. Given a set $A=\{\eta_{1},\cdots,\eta_{d}\}$ of $d$ linearly independent vectors in ${\mathbb{R}}^{d}$, the convex hull $T=\text{conv}_{{\mathbb{R}}^{d}}(A)$ is called a surface simplex spanned by the set $A$.

The following lemma can be verified by straightforward calculations.

Lemma 2.2**.**

Let $T:=\text{conv}_{{\mathbb{R}}^{d}}\{\eta_{1},\cdots,\eta_{d}\}$ denote a surface simplex in ${\mathbb{R}}^{d}$ and $S=\text{conv}_{\mathbb{S}^{d-1}}\{\eta_{1},\cdots,\eta_{d}\}$ a geodesic simplex in $\mathbb{S}^{d-1}$ both being spanned by a set of linearly independent vectors $\eta_{1},\cdots,\eta_{d}\in\mathbb{S}^{d-1}$. Let $H_{T}$ denote the hyperplane passing through the points $\eta_{1},\cdots,\eta_{d}$. Let $g(x)=\frac{x}{\|x\|}$ for $x\in{\mathbb{R}}^{d}\setminus\{0\}$. Then the following statements hold:

1. (i)

$x\in S$* if and only if $x\in\mathbb{S}^{d-1}$ and $x=\sum_{j=1}^{d}t_{j}\eta_{j}$ for some $t_{1},\cdots,t_{d}\geq 0$.* 2. (ii)

$g:x\mapsto\frac{x}{\|x\|}$* is a bijective continuously differentiable mapping from $T$ onto $S$ that maps each convex subset of $T$ to a geodesically convex subset of $S$. Furthermore, $g([x,y])=\operatorname{Arc}(g(x),g(y))$ for any two distinct points $x,y\in T$.* 3. (iii)

For each nonnegative measurable function $F:S\to[0,\infty)$,

[TABLE]

where $a:=\min_{x\in T}\|x\|$. 4. (iv)

If $\min_{x\in T}\|x\|=a$, then

[TABLE]

The following geometric fact, which can also be easily verified through straightforward calculations, will be used frequently in our proof.

Lemma 2.3**.**

Assume that $x=\xi\cos{\theta}_{1}+\eta_{1}\sin{\theta}_{1}$ and $y=\xi\cos{\theta}_{2}+\eta_{2}\sin{\theta}_{2}$, where $\xi,\eta_{1},\eta_{2}\in\mathbb{S}^{d-1}$, $\eta_{i}\cdot\xi=0$ and $0\leq{\theta}_{i}\leq\pi$ for $i=1,2$. Then

[TABLE]

Thus,

[TABLE]

Finally, we need the following geometric property concerning convex sets in ${\mathbb{R}}^{d}$.

Lemma 2.4**.**

If $G$ is a convex subset of ${\mathbb{R}}^{d}$ with diameter $2r$ and volume $\geq c_{d}r^{d-1}a$ for some $a\in(0,r]$, then $G$ contains a Euclidean ball of radius $c_{d}^{\prime}a$ for some positive constant $c_{d}^{\prime}$.

Proof.

We use induction on the dimension $d$. The conclusion holds trivially for $d=1$ since every convex subset of ${\mathbb{R}}$ must be an interval. Now assume that the conclusion has been proven in ${\mathbb{R}}^{d-1}$, and we will deduce it for the case of ${\mathbb{R}}^{d}$ as follows. We denote by $m_{d}$ the $d$-dimensional Lebesgue measure (i.e., the $d$-dimensional Hausdorff measure). Let $p,q\in\overline{G}$ be such that $2r=|p-q|=\text{diam}(G)$, and let $\xi$ denote the unit vector in the direction of $q-p$. Then $\overline{G}\subset\{x\in{\mathbb{R}}^{d}:\ \ 0\leq(x-p)\cdot\xi\leq 2r\}.$ For $0\leq t\leq 2r$, let $G(t)=\{x\in\overline{G}:\ \ (x-p)\cdot\xi=t\}$. It is easily seen that each slice $G(t)$ is a compact convex subset of the hyperplane $S(t):=\{x\in{\mathbb{R}}^{d}:\ \ (x-p)\cdot\xi=t\}$, and moreover,

[TABLE]

Thus, there must exist $0<t_{0}<2r$ such that $m_{d-1}(G(t_{0}))>c^{\prime}r^{d-2}a>0.$ Since $\text{diam}(G(t_{0}))\leq\text{diam}(G)=2r$, it follows by the induction hypothesis that $G(t_{0})$ contains a $(d-1)$-dimensional ball $B=\{x\in S(t_{0}):\ \ |x-z_{0}|\leq ca\}$ for some $z_{0}\in S(t_{0})$ and $c=c_{d}>0$. Without loss of generality, we may assume that $r\leq t_{0}\leq 2r$ since otherwise we interchange the order of the points $p$ and $q$. It suffices to show that the convex hull $H$ of the set $B\cup\{p\}$ contains a ball of radius $c_{d}a$. Indeed, by rotation invariance, we may assume that $p=0$ and $\xi=e_{d}$. We then write $z_{0}=(u_{0},t_{0})$ and let $Q$ denote the cube in ${\mathbb{R}}^{d-1}$ centered at $u_{0}$ and having side length $\varepsilon_{d}a$ for a sufficiently small constant $\varepsilon_{d}$. Clearly, the rectangle $R=Q\times[(1-\frac{\varepsilon_{d}a}{r})t_{0},t_{0}]$ contains a $d$-dimensional ball of radius $c^{\prime}_{d}a$. However, a straightforward calculation shows that $R\subset H$. ∎

2.2. Organization of the proof

We divide the proof into two main steps. At the first step, we prove

Proposition 2.5**.**

Let $w$ be a weight on $\mathbb{S}^{d-1}$ normalized by $w(\mathbb{S}^{d-1})=1$. If $\frac{N}{2^{d-2}}\in{\mathbb{N}}$ and $N\geq\max_{x\in\mathbb{S}^{d-1}}\frac{1}{w(B(x,10^{-d}))}$, then there exists a partition $\{T_{1},\cdots,T_{m}\}$ of $\mathbb{S}^{d-1}$ such that each $T_{j}$ is an admissible geodesic simplex in $\mathbb{S}^{d-1}$ satisfying $Nw(T_{j})\in{\mathbb{N}}$.

To state our main result in the next step, we need to introduce some notation. Given a surface simplex $T$ in ${\mathbb{R}}^{d}$, we denote by $H_{T}$ the hyperplane in which the surface simplex $T$ lies, and $B(x,r)_{H_{T}}$ the ball $\{y\in H_{T}:\ \ \|x-y\|\leq r\}$ with center $x\in H_{T}$ and radius $r>0$ in the hyperplane $H_{T}$. For a weight function $w$ on $T$, we write $w(E)=\int_{E}w(x)\,dx$ for $E\subset T$, where $dx$ denotes the surface Lebesgue measure on $T$.

At the second step, we prove

Proposition 2.6**.**

Let $T$ be a surface simplex in ${\mathbb{R}}^{d}$ spanned by a admissible subset of $\mathbb{S}^{d-1}$ with $d\geq 3$, and let $w$ be a weight on $T$ such that $Nw(T)\in{\mathbb{N}}$ for some $N\in{\mathbb{N}}$. Assume that there exists $r_{N}\in(0,1)$ such that $w(B)\geq\frac{1}{N}$ for every ball $B=B(x,r_{N})_{H_{T}}\subset T$. Then there exists a convex partition $\{E_{1},\cdots,E_{n_{0}}\}$ of $T$ such that $Nw(E_{j})\in{\mathbb{N}}$ and $B(x_{j},c_{d}^{\prime}r_{N})_{H_{T}}\subset E_{j}\subset B(x_{j},c_{d}r_{N})_{H_{T}}$ for some $x_{j}\in E_{j}$ and all $1\leq j\leq n_{0}$.

For the moment, we take Proposition 2.5 and Proposition 2.5 for granted and proceed with the proof of Theorem 1.4.

Proof of Theorem 1.4.

Assume that $\frac{N}{2^{d-2}}\in{\mathbb{N}}$, and set ${\delta}_{0}=\min_{x\in\mathbb{S}^{d-1}}w(B(x,10^{-d}))$. We consider the following two cases:

Case 1:    $1\leq N\leq{\delta}_{0}^{-1}$.

In this case, $r\geq 10^{-d}$. Thus, by Lemma 2.2 and Lemma 2.4, it suffices to prove the following assertion: given any integer $M\geq 1$ and any weight $w$ such that $w(B)>0$ for every spherical cap $B\subset\mathbb{S}^{d-1}$, there exists a convex partition $R_{1},\cdots,R_{M}$ of $\mathbb{S}^{d-1}$ with $w(R_{j})=\frac{1}{M}$ for $1\leq j\leq M$. We prove this last assertion by induction on the dimension $d$. If $d=2$, then the stated assertion follows directly by continuity. Now assume that the assertion holds on the sphere $\SS^{d-2}$ with $d\geq 3$. Let $w_{d-1}(\xi)=\int_{0}^{\pi}w(\cos{\theta},\xi\sin{\theta})\sin^{d-2}{\theta}\,d{\theta}$ for $\xi\in\SS^{d-2}$. Clearly, $w_{d-1}(B)>0$ for every spherical cap $B$ in $\SS^{d-2}$. By the induction hypothesis, there exists a convex partition $E_{1},\cdots,E_{M}$ of $\SS^{d-2}$ with $w_{d-1}(E_{j})=\frac{1}{M}$ for $1\leq j\leq M$. Now set

[TABLE]

It is easily seen that $\{R_{1},\cdots,R_{M}\}$ is a convex partition of $\mathbb{S}^{d-1}$. Moreover, for each $1\leq j\leq M$,

[TABLE]

This completes the induction.

Case 2. $N\geq{\delta}_{0}^{-1}$.

In this case, $w(B(x,10^{-d}))\geq{\delta}_{0}\geq\frac{1}{N}$ for any $x\in\mathbb{S}^{d-1}$. Hence, applying Proposition 2.5, there exists a partition $\mathbb{S}^{d-1}=\bigcup_{j=1}^{m}S_{j}$ of $\mathbb{S}^{d-1}$ such that each $S_{j}$ is an admissible geodesic simplex in $\mathbb{S}^{d-1}$ with $Nw(S_{j})\in{\mathbb{N}}$. Let $T_{j}$ be the surface simplex in ${\mathbb{R}}^{d}$ such that $g(T_{j})=S_{j}$. Set $a_{j}:=\min_{x\in T_{j}}\|x\|$ and define $w_{j}(x)=w(g(x))\frac{a_{j}}{\|x\|^{d}}$ for $x\in T_{j}$. It’s easily seen that $\frac{1}{2d}\leq a_{j}\leq 1$. Moreover, according to Lemma 2.2 (iii) and (iv), $w_{j}$ is a weight on $T_{j}$ satisfying the following two conditions: (a) $w_{j}(E)=w(g(E))$ for each $E\subset T_{j}$, and (b) there exists a constant $r_{N}\sim_{d}r$ such that

[TABLE]

whenever $B(x,r_{N})_{H_{T_{j}}}\subset T_{j}$. Thus, applying Proposition 2.6 to the weight $w_{j}$ on each surface simplex $T_{j}$, we obtain a convex partition $T_{j}=\bigcup_{i=1}^{n_{j}}E_{j,i}$ such that $B(x_{j,i},c_{1}r_{N})_{H_{T_{j}}}\subset E_{j,i}\subset B(x_{j,i},c_{2}r_{N})_{H_{T_{j}}}$ and $Nw_{j}(E_{j,i})\in{\mathbb{N}}$ for all $1\leq i\leq n_{j}$. Finally, setting $R_{j,i}=g(E_{j,i})$ for $1\leq j\leq m$ and $1\leq i\leq n_{j}$, and applying Lemma 2.2 (ii), we obtain a convex partition $\{R_{j,i}:\ \ 1\leq j\leq m,\ 1\leq i\leq n_{j}\}$ of $\mathbb{S}^{d-1}$ such that $Nw(R_{j,i})\in{\mathbb{N}}$ and $B(g(x_{j,i}),c_{1}^{\prime}r_{N})\subset R_{j,i}\subset B(g(x_{j,i}),c_{2}^{\prime}r_{N})$ for all $1\leq i\leq n_{j}$ and $1\leq j\leq m$.

∎

2.3. Proof of Proposition 2.5

For simplicity, we say that a set $E\subset\mathbb{S}^{d-1}$ has a regular geodesic simplex partition with respect to the integer $N$ and the weight $w$ on $\mathbb{S}^{d-1}$ if it can be written as a finite union of admissible geodesic simplexes $S_{1},\cdots,S_{n}$ whose interiors are pairwise disjoint and such that $Nw(S_{j}))\in{\mathbb{N}}$ for each $1\leq j\leq n$.

First, we claim that Proposition 2.5 is a consequence of the following assertion:

(A)     *The spherical cap $B(e_{d},\frac{\pi}{2})\subset\mathbb{S}^{d-1}$ has a regular geodesic simplex partition with respect to the integer $N$ and the weight $w$ on $\mathbb{S}^{d-1}$ whenever $N$ and $w$ satisfy the following conditions (i) $w(B(e_{d},\frac{\pi}{2}))=\frac{1}{2}$; (ii) $\frac{N}{2^{d-2}}\in{\mathbb{N}}$; and (iii) $N\geq\max_{x\in\mathbb{S}^{d-1}}\frac{1}{w(B(x,10^{-d}))}$, where $e_{d}=(0,0,\cdots,0,1)\in\mathbb{S}^{d-1}$.

To see this, let $w$ be a weight on $\mathbb{S}^{d-1}$ satisfying the conditions of Proposition 2.5. Then $w(B(x,\frac{\pi}{2}))+w(B(-x,\frac{\pi}{2}))=w(\mathbb{S}^{d-1})=1$ for all $x\in\mathbb{S}^{d-1}$. It follows by continuity that $w(B(x_{0},\frac{\pi}{2}))=w(B(-x_{0},\frac{\pi}{2}))=\frac{1}{2}$ for some $x_{0}\in\mathbb{S}^{d-1}$. Let $\rho\in SO(d)$ be a rotation such that $\rho e_{d}=x_{0}$ and set $\widetilde{w}(x):=w(\rho x)$ for $x\in\mathbb{S}^{d-1}$. Applying Assertion (A) to the weight $\widetilde{w}$ on $\mathbb{S}^{d-1}$ and the integer $N$, we conclude that the spherical cap $B(x_{0},\frac{\pi}{2})$ has a regular geodesic simplex partition with respect to $N$ and $w$. A similar argument also yields the same conclusion for the spherical cap $B(-x_{0},\frac{\pi}{2})$. Since $\{B(x_{0},\frac{\pi}{2}),B(-x_{0},\frac{\pi}{2})\}$ is a partition of $\mathbb{S}^{d-1}$, it follows that the sphere $\mathbb{S}^{d-1}$ itself has a regular geodesic simplex partition with respect to $N$ and $w$. This shows the claim.

Assertion A can be proved using induction on the dimension $d$. We start with the case of $d=3$. For simplicity, we write $\xi_{\varphi}:=(\cos\varphi,\sin\varphi,0)$ for $0\leq\varphi\leq 2\pi$, and set $T({\alpha},{\beta}):=\text{conv}_{\SS^{2}}\{e_{3},\xi_{\alpha},\xi_{\beta}\}$ for $0\leq{\alpha}<{\beta}\leq 2\pi$; that is,

[TABLE]

By Lemma 2.3, it is easily seen that for any $\varphi\in[0,2\pi)$,

[TABLE]

which in particular implies that

[TABLE]

By continuity, we conclude that for any $\varphi\in[0,2\pi]$, there exists ${\alpha}\in[\varphi+\frac{\pi}{6},\varphi+\frac{\pi}{3}]$ such that Nw\Bigl{(}T(\varphi,{\alpha})\Bigr{)}\in{\mathbb{N}}. Invoking this fact iteratively, we obtain a sequence of numbers $\{{\alpha}_{j}\}_{j=0}^{k_{0}-1}$ such that ${\alpha}_{0}=0$,

[TABLE]

where $k_{0}\leq 12$ is the smallest positive integer such that ${\alpha}_{k_{0}-1}+\frac{\pi}{6}<2\pi$. Setting ${\alpha}_{k_{0}}=2\pi$, we have that $\frac{\pi}{6}<2\pi-{\alpha}_{k_{0}-1}={\alpha}_{k_{0}}-{\alpha}_{k_{0}-1}\leq\frac{\pi}{2}$. Since $N$ is even, we have

[TABLE]

Thus, Nw\bigl{(}T({\alpha}_{k_{0}-1},{\alpha}_{k_{0}})\bigr{)}\in{\mathbb{N}}. Since $B(e_{3},\frac{\pi}{2})=\bigcup_{j=1}^{k_{0}}T({\alpha}_{j-1},{\alpha}_{j})$, we prove Assertion (A) for the case of $d=3$.

Now assume that Assertion A holds on the sphere $\mathbb{S}^{d-1}$ for some $d\geq 3$, which in turn implies Proposition 2.5 for the sphere $\mathbb{S}^{d-1}$. To show Assertion (A) on $\SS^{d}$, assume that $\frac{N}{2^{d-1}}\in{\mathbb{N}}$ and let $w_{d}$ be a weight on $\SS^{d}$ such that $w_{d}(B(e_{d+1},\frac{\pi}{2}))=\frac{1}{2}$ and $N\geq\max_{x\in\mathbb{S}^{d-1}}\frac{1}{w(B(x,10^{-d-1}))}$. For $E\subset\mathbb{S}^{d-1}$, we write,

[TABLE]

Define

[TABLE]

Clearly, $w_{d-1}$ is a weight on the sphere $\mathbb{S}^{d-1}$ satisfying that for each $E\subset\mathbb{S}^{d-1}$,

[TABLE]

In particular, w_{d-1}(\mathbb{S}^{d-1})=2w_{d}\Bigl{(}B_{\SS^{d}}(e_{d+1},\frac{\pi}{2})\Bigr{)}=1. Next, setting ${\delta}_{d}=10^{-d-1}$ and $N_{1}=N/2$, we claim that

[TABLE]

where the notation $B_{\SS^{\ell}}(x,r)$ is used for the spherical cap in $\SS^{\ell}$. For the proof of (2.4), it is sufficient to show that for any $x\in\mathbb{S}^{d-1}$,

[TABLE]

$y_{x}=(x\sin\frac{\pi}{4},\cos\frac{\pi}{4})\in\SS^{d}$. Indeed, once (2.5) is proved, then

[TABLE]

To show (2.5), let $z=(\xi\sin{\theta},\cos{\theta})\in B_{\SS^{d}}(y_{x},{\delta}_{d})$ with ${\theta}\in[0,\pi]$ and $\xi\in\mathbb{S}^{d-1}$. Then by Lemma 2.3, $|{\theta}-\frac{\pi}{4}|\leq\mathtt{d}(y_{x},z)\leq{\delta}_{d}$ (which implies $\frac{\pi}{6}\leq{\theta}\leq\frac{\pi}{3}$), and

[TABLE]

This means that $\xi\in B_{\mathbb{S}^{d-1}}(x,{\delta}_{d-1})$ and hence z=(\xi\sin{\theta},\cos{\theta})\in S\Bigl{(}B_{\mathbb{S}^{d-1}}(x,{\delta}_{d-1})\Bigr{)}. (2.5) then follows.

Now applying the induction hypothesis to the weight $w_{d-1}$ on $\mathbb{S}^{d-1}$ and the integer $N_{1}$, we conclude that there exists a regular geodesic simplex partition $\{E_{1},\cdots,E_{m}\}$ of $\mathbb{S}^{d-1}$ with respect to the integer $N_{1}$ and the weight $w_{d-1}$ on $\mathbb{S}^{d-1}$. Setting $S_{j}=S(E_{j})$ for $1\leq j\leq m$, we obtain a partition $\{S_{1},\cdots,S_{m}\}$ of the spherical cap $B_{\SS^{d}}(e_{d+1},\frac{\pi}{2})$ satisfying that for each $1\leq j\leq m$,

[TABLE]

Thus, to complete the proof, it remains to show that each $S_{j}$ is an admissible geodesic simplex on $\SS^{d}$. Since each $E_{j}$ is an admissible geodesic simplex in $\mathbb{S}^{d-1}$, there exists an admissible subset $\{\xi_{j,1},\cdots,\xi_{j,d}\}$ of $\mathbb{S}^{d-1}$ such that $E_{j}=\text{conv}_{\mathbb{S}^{d-1}}\{\xi_{j,1},\cdots,\xi_{j,d}\}$ for each $1\leq j\leq m$. Using Lemma 2.2 (i), it is easily seen that

[TABLE]

where $\widehat{\xi_{j,i}}=(\xi_{j,i},0)\in\SS^{d}$ for $i=1,\cdots,d$. Since $\{\widehat{\xi_{j,1}},\cdots,\widehat{\xi_{j,d}},e_{d+1}\}$ is an admissible subset of $\SS^{d}$, we conclude that each $S_{j}$ is an admissible geodesic simplex of $\SS^{d}$. This completes the proof.

Remark 2.7*.*

If $w$ is a $\tau_{d}$-invariant weight on $\mathbb{S}^{d-1}$ with $w(\mathbb{S}^{d-1})=1$, then the condition (i) of Assertion (A) always holds.

Remark 2.8*.*

If $w$ is a normalized ${\mathbb{Z}}_{2}^{d}$-invariant weight on $\mathbb{S}^{d-1}$, and $S=\text{conv}_{\mathbb{S}^{d-1}}\{e_{1},\cdots,e_{d}\}$, then $S_{\tau}:=\{x\tau:\ \ x\in S\}$ is an admissible geodesic simplex with $w(S_{\tau})=\frac{1}{2^{d}}$ for each $\tau\in{\mathbb{Z}}_{2}^{d}$. Moreover, $\{S_{\tau}:\ \ \tau\in{\mathbb{Z}}_{2}^{d}\}$ is a partition of $\mathbb{S}^{d-1}$. Thus, Proposition  2.5 holds trivially if $w$ is ${\mathbb{Z}}_{2}^{d}$-invariant, $w(\mathbb{S}^{d-1})=1$ and $\frac{N}{2^{d}}\in{\mathbb{N}}$.

2.4. Proof of Proposition 2.6

Let $e_{1}=(1,0,\cdots,0),\cdots,e_{d}=(0,\cdots,0,1)$ denote the standard canonical basis in ${\mathbb{R}}^{d}$. Without loss of generality, we may assume that $T=\text{conv}_{{\mathbb{R}}^{d}}\{\xi_{1},e_{2},\cdots,e_{d}\}$ with $\xi_{1}=(\sin{\theta},\cos{\theta},0,\cdots,0)$ for some $\frac{\pi}{6}\leq{\theta}\leq\frac{\pi}{2}$, in which case $T$ can be written explicitly as

[TABLE]

For the rest, we fix ${\theta}\in[\frac{\pi}{6},\frac{\pi}{2}]$, and denote by $T_{d}$ the simplex in ${\mathbb{R}}^{d}$ given by

[TABLE]

Also, we will use the notation $B(x,r)_{{\mathbb{R}}^{d}}$ to denote the Euclidean ball $\{y\in{\mathbb{R}}^{d}:\ \ \|y-x\|\leq r\}$ in ${\mathbb{R}}^{d}$.

For the proof of Proposition 2.6, we claim that it is sufficient to show the following assertion:

(B) *If $d\geq 2$, $N\in{\mathbb{N}}$, $r\in(0,1)$ and $w$ is a weight on $T_{d}$ such that $Nw(T_{d})\in{\mathbb{N}}$, and \inf\bigl{\{}w(B):B=B(x,r)_{{\mathbb{R}}^{d}}\subset T_{d}\bigr{\}}\geq\frac{1}{N}, then $T_{d}$ has a convex partition $\{R_{1},\cdots,R_{n_{0}}\}$ which satisfies that $Nw(R_{j})\in{\mathbb{N}}$ and $B(x_{j},c_{d}^{\prime}r)_{{\mathbb{R}}^{d}}\subset R_{j}\subset B(x_{j},c_{d}r)_{{\mathbb{R}}^{d}}$ for each $1\leq j\leq n_{0}$ and some constants $c_{d}^{\prime},c_{d}>0$.

To show the claim, for $d\geq 3$, we let $\Psi:T_{d-1}\to T$ denote the mapping given by $x\mapsto(x,\varphi(x))$ with $\varphi(x)=1-x_{1}\tan\frac{{\theta}}{2}-\sum_{j=2}^{d-1}x_{j}$ for $x\in T_{d-1}$. Clearly, $\Psi:T_{d-1}\to T$ is a bijective mapping that maps convex sets to convex sets and satisfies $\|x-y\|\leq\|\Psi(x)-\Psi(y)\|\leq C_{d}\|x-y\|$ for any $x,y\in T_{d-1}$. Moreover, for each nonnegative function $f:T\to[0,\infty)$, we have

[TABLE]

where $a_{\theta}:=\sqrt{d-1+\tan^{2}\frac{{\theta}}{2}}$. Thus, setting $w_{d-1}(x)=a_{\theta}w(\Psi(x))$ for $x\in T_{d-1}$, we have that $w_{d-1}(E)=w(\Psi(E))$ for each $E\subset T_{d-1}$. In particular, this implies that $Nw_{d-1}(T_{d-1})=Nw(T)\in{\mathbb{N}}$ and $w_{d-1}(B)=w(\Psi(B))\geq\frac{1}{N}$ whenever $B=B(x,r_{N})_{{\mathbb{R}}^{d-1}}\subset T_{d-1}$. Thus, applying Assertion (B) to the integer $N$ and the weight $w_{d-1}$ on $T_{d-1}$, we get a partition $\{R_{1},\cdots,R_{n_{0}}\}$ of $T_{d-1}$. It follows that $\{E_{j}=\Psi(R_{j}):\ \ j=1,\cdots,n_{0}\}$ is a convex partition of the surface simplex $T$ with the stated properties in Proposition 2.6. This shows the claim.

It remains to prove Assertion  (B). We shall use induction on the dimension $d$. Let $L_{d}\geq 10$ be a large constant that will be specified later. Let $r_{1}=L_{d}r$. Without loss of generality, we may assume that $0<r\leq\frac{1}{100L_{d}}$ since otherwise Assertion (B) holds trivially with $n_{0}=1$.

We start with the case of $d=2$. Note that $T_{2}$ is an isosceles triangle with vertices at $(0,0)$, $(0,1)$ and $(\sin{\theta},\cos{\theta})$ in the $x_{1}x_{2}$-plane. By rotation invariance of Assertion (B), we may assume, without loss of generality, that $T_{2}$ is the triangle with vertices at $A=(-\sin\frac{{\theta}}{2},\cos\frac{{\theta}}{2})$, $B=(\sin\frac{{\theta}}{2},\cos\frac{{\theta}}{2})$ and $O=(0,0)$ in the $x_{1}x_{2}$-plane. For $0\leq t<s\leq 1$, we set ${\triangle}(t,s):=\{(ux,uy):\ \ (x,y)\in T_{2},\ \ t\leq u\leq s\}$. It is easily seen that for any $0\leq s\leq 1-3r_{1}$, the set ${\triangle}(s+r_{1},s+3r_{1})$ contains a ball of radius $r$, and hence $w({\triangle}(s+r_{1},s+3r_{1}))\geq\frac{1}{N}$. By continuity, this implies that for each $s\in[0,1-3r_{1}]$, there exists $t\in[s+r_{1},s+3r_{1}]$ such that $Nw({\triangle}(s,t))\in{\mathbb{N}}$. Using this fact iteratively, we can construct a partition $0=t_{0}<t_{1}<\cdots<t_{k}=1$ of $[0,1]$ such that $t_{j-1}+r_{1}\leq t_{j}\leq t_{j-1}+3r_{1}$ and $Nw({\triangle}(t_{j-1},t_{j}))\in{\mathbb{N}}$ for $1\leq j\leq k-1$, where $k$ is the largest positive integer so that $3r_{1}\leq 1-t_{k-1}\leq 6r_{1}$. Since $Nw(T_{2})\in{\mathbb{N}}$, we also have that $Nw({\triangle}(t_{k-1},1))\in{\mathbb{N}}$. Note that for $0\leq j\leq k-1$,

[TABLE]

For simplicity, we set ${\triangle}_{j}:={\triangle}(t_{j-1},t_{j})$ for $1\leq j\leq k$. We construct a convex partition of the domain ${\triangle}_{j}$ for each $10\leq j\leq k$ as follows. Let $A_{j}(s)=t_{j}A+t_{j}(B-A)s$, $s\in[0,1]$ denote the parametric representation of the line segment from $t_{j}A$ to $t_{j}B$. For $0\leq s<t\leq 1$, we denote by $T_{j}(s,t)$ the trapezoid with vertices at $A_{j-1}(s)$, $A_{j-1}(t)$, $A_{j}(s)$ and $A_{j}(t)$. It follows from (2.6) that for $0\leq s\leq 1-j^{-1}$

[TABLE]

and $\text{diam}(T_{j}(s,s+j^{-1}))\sim r_{1}$ . By Lemma 2.4, this implies that $T_{j}(s,s+j^{-1})$ contains a ball of radius $c_{2}L_{2}r$. Thus, taking $L_{2}>c_{2}^{-1}$, we conclude that w\Bigl{(}T_{j}(s,s+j^{-1})\Bigr{)}\geq\frac{1}{N} for all $s\in[0,1-j^{-1}]$. By continuity, this implies that for any $s\in[0,1-3j^{-1}]$, there exists $t\in[s+j^{-1},s+3j^{-1}]$ such that $Nw(T_{j}(s,t))\in{\mathbb{N}}$. Applying this fact iteratively, and recalling that $Nw({\triangle}_{j})\in{\mathbb{N}}$, we may construct a partition $0=s_{j,0}<s_{j,1}<\cdots<s_{j,m_{j}}=1$ of $[0,1]$ such that $s_{j,i-1}+j^{-1}\leq s_{j,i}\leq s_{j,i}+3j^{-1}$ for $1\leq i\leq m_{j}-1$, $j^{-1}\leq s_{j,m_{j}}-s_{j,m_{j}-1}\leq 6j^{-1}$ and Nw\Bigl{(}T_{j}(s_{j,i-1},s_{j,i})\Bigr{)}\in{\mathbb{N}} for $1\leq i\leq m_{j}$. Using Lemma 2.4, we know that each set $T_{j}(s_{j,i-1},s_{j,i})$ contains a ball of radius $r$ and has diameter $\sim r$.

Now putting the above together, we obtain a convex partition of $T_{2}$:

[TABLE]

Assertion (B) for $d=2$ then follows.

Now assume that Assertion (B) has been proven for the simplex $T_{d-1}\subset{\mathbb{R}}^{d-1}$. Let $N$ and $w$ be a positive integer and a weight on $T_{d}$ satisfying the conditions of Assertion (B). Note that for any nonnegative function $f$ on $T_{d}$,

[TABLE]

For $0\leq t<1-r_{1}$, define

[TABLE]

It is easily seen that $E_{t}$ is a convex set in ${\mathbb{R}}^{d}$ with diameter $\sim t+r_{1}$. Furthermore, by (2.7), we have $\text{Vol}_{d}(E_{t})\sim(t+r_{1})^{d-1}r_{1}$. Thus, according to Lemma 2.4, for each $0\leq t<1-r_{1}$, the set $E_{t}$ must contain a ball of radius at least $c_{d}r_{1}=c_{d}L_{d}r$. Taking the constant $L_{d}$ sufficiently large so that $c_{d}L_{d}>1$, we obtain from the assumption of Assertion (B) that that $w(E_{t})\geq\frac{1}{N}$ for all $0\leq t\leq 1-r_{1}$. This together with the fact that $Nw(T_{d})\in{\mathbb{N}}$ implies that there is a partition $0=t_{0}<t_{1}<\cdots<t_{m}=1$ of $[0,1]$ such that $t_{j-1}+r_{1}\leq t_{j}\leq t_{j-1}+3r_{1}$ for $1\leq j\leq m-1$, $t_{m-1}+r_{1}\leq t_{m}\leq t_{m-1}+3r_{1}$ and $Nw(A_{j})\in{\mathbb{N}}$ for $1\leq j\leq m$, where A_{j}:=\Bigl{\{}(sx^{\prime},1-s):x^{\prime}\in T_{d-1},\ \ t_{j-1}\leq s\leq t_{j}\Bigr{\}}. Next, for each $1\leq j\leq m$, we define a weight function $w_{j}$ on $T_{d-1}$ by

[TABLE]

We claim that for each ball $B\subset T_{d-1}$ with radius $r_{1}$, $w_{j}(B)\geq\frac{1}{N}$. Indeed, setting

[TABLE]

and using (2.7), we see that $\widetilde{B}_{j}$ is a convex subset of $T_{d}$ with diameter $\sim r_{1}$ and volume $\sim r_{1}^{d}$, By Lemma 2.4, this implies that $\widetilde{B}_{j}$ contains a ball of radius $c_{d}r_{1}\geq r$. Using (2.7), we obtain $w_{j}(B)=w(\widetilde{B}_{j})\geq\frac{1}{N}.$ This shows the claim.

Now applying the induction hypothesis to the integer $N$, the radius $r_{1}$ and the weights $w_{j}$ on $T_{d-1}$, we conclude that for each $1\leq j\leq m$, the simplex $T_{d-1}$ has a convex partition $R_{j,1},\cdots,R_{j,k_{j}}$ such that $B(x_{j,i},c_{d}r_{1})_{{\mathbb{R}}^{d-1}}\subset R_{j,i}\subset B(x_{j,i},c_{d}^{\prime}r_{1})_{{\mathbb{R}}^{d-1}}$ and $Nw_{j}(R_{j,i})\in{\mathbb{N}}$. Setting

[TABLE]

we get a convex partition $A_{j}=\bigcup_{i=1}^{k_{j}}T_{j,i}$ of the set $A_{j}$ satisfying that $Nw(T_{j,k})=Nw_{1}(R_{j,k})\in{\mathbb{N}}$, $\text{diam}(T_{j,k})\sim r_{1}$ and $\text{Vol}_{d}(T_{j,k})\sim r_{1}^{d}$. This shows that $\{T_{j,i}:\ \ 1\leq j\leq m,\ \ 1\leq i\leq k_{j}\}$ is a convex partition of $T_{d}$ with the stated properties in Assertion (B).

2.5. Concluding remarks

Combining Remarks 2.7 and 2.8 with Proposition 2.6, we obtain

Corollary 2.9**.**

Let $w$ be a weight on $\mathbb{S}^{d-1}$ that is either $\tau_{d}$-invariant or ${\mathbb{Z}}_{2}^{d}$-invariant and which satisfies the conditions of Theorem 1.4 for some $r\in(0,\pi)$ and positive even integer $N$. Assume in addition that $2^{-d}N$ is an integer if $w$ is ${\mathbb{Z}}_{2}^{d}$-invariant. Let $S$ denote either the semisphere $\{x\in\mathbb{S}^{d-1}:\ \ x_{d}\geq 0\}$ or the geodesic simplex $\{x\in\mathbb{S}^{d-1}:\ \ x_{1},\cdots,x_{d}\geq 0\}$ according to whether $w$ is $\tau_{d}$-invariant or ${\mathbb{Z}}_{2}^{d}$-invariant. Then $S$ has a convex partition $\{R_{1},\cdots,R_{M}\}$ for which the sets $R_{j}\subset S$ satisfy Conditions (i) and (ii) of Theorem 1.4.

3. Proof of Theorem 1.4: the general case of $N\in{\mathbb{N}}$

Our goal in this section is to prove Theorem 1.4 without the extra condition $\frac{N}{2^{d-2}}\in{\mathbb{N}}$. Assume that $w$ is a weight on $\mathbb{S}^{d-1}$ normalized by $w(\mathbb{S}^{d-1})=1$ and satisfying that $w(B)>0$ for every spherical cap $B\subset\mathbb{S}^{d-1}$. Let $r\in(0,1)$ and a positive integer $N$ satisfying that $\inf_{x\in\mathbb{S}^{d-1}}w(B(x,r))\geq\frac{1}{N}$. We will keep these assumptions throughout this section. For convenience, we introduce the following concept.

Definition 3.1**.**

Given a parameter $\varepsilon\in(0,1)$, we say a subset $A=\{\eta_{1},\cdots,\eta_{m}\}$ of $m$ distinct points ($m\leq d$) on $\mathbb{S}^{d-1}$ is strongly $\varepsilon$-separated if for each $j=1,\cdots,m$,

[TABLE]

A geodesic simplex in $\mathbb{S}^{d-1}$ is said to be in the class $\mathcal{S}_{\varepsilon}$ if it is spanned by a set of $d$ strongly $\varepsilon$-separated points $\eta_{1},\cdots,\eta_{d}$ in $\mathbb{S}^{d-1}$.

Theorem 1.4 is a direct consequence of the following two propositions.

Proposition 3.2**.**

Let $\varepsilon\in(0,1)$ be a given parameter and $S$ a geodesic simplex from the class $\mathcal{S}_{\varepsilon}$. If $Nw(S)\in{\mathbb{N}}$, then there exists a convex partition $\{R_{1},\cdots,R_{n}\}$ of $S$ such that $Nw(R_{j})\in{\mathbb{N}}$ and $B_{\mathbb{S}^{d-1}}(x_{j},c_{d,\varepsilon}r)\subset R_{j}\subset B_{\mathbb{S}^{d-1}}(x_{j},c_{d,\varepsilon}^{\prime}r)$ for each $1\leq j\leq n$.

Proposition 3.3**.**

There exists a partition $\{S_{1},\cdots,S_{m}\}$ of $\mathbb{S}^{d-1}$ such that each $S_{j}$ is a geodesic simplex from the class $\mathcal{S}_{\varepsilon_{d}}$ satisfying that $Nw(S_{j})\in{\mathbb{N}}$, where $\varepsilon_{d}$ is a positive parameter depending only on $d$.

Proposition 3.2 is a direct consequence of Proposition 2.6. For completeness, we write its proof below.

Proof of Proposition 3.2.

Firstly, assume that $S$ is spanned by a set of strongly $\varepsilon$-separated points $\eta_{1},\cdots,\eta_{d}\in\mathbb{S}^{d-1}$ for some $\varepsilon\in(0,1)$, and let $T=\text{conv}_{{\mathbb{R}}^{d}}\{\eta_{1},\cdots,\eta_{d}\}$. We claim that $a:=\min_{x\in T}\|x\|\geq\varepsilon/d$. To see this, let $x=\sum_{j=1}^{d}t_{j}\eta_{j}\in T$ be such that $a=\|x\|$, where $t_{1},\cdots,t_{d}\geq 0$ and $\sum_{j=1}^{d}t_{j}=1$. Without loss of generality, we may assume that $t_{1}=\max_{1\leq j\leq d}t_{j}$. Then $t_{1}\geq\frac{1}{d}$, and

[TABLE]

Secondly, let $A:=[\eta_{1}\ \eta_{2}\ \cdots\ \eta_{d}]$ denote the nonsingular $d\times d$ real matrix with column vectors $\eta_{1},\cdots,\eta_{d}$. We assert that

[TABLE]

Without loss of generality, we may assume that $x\in\mathbb{S}^{d-1}$ and $|x_{1}|=\max_{1\leq j\leq d}|x_{j}|$. On the one hand, by the Cauchy-Schwarz inequality,

[TABLE]

On the other hand, however, since $|x_{1}|\geq\frac{1}{\sqrt{d}}$, we have

[TABLE]

This proves the inequality (3.1).

Finally, we prove the assertion of Proposition 3.2. Consider the bijective mapping $\Phi:x\mapsto g(Ax)$ from the surface simplex $T_{0}=\text{conv}_{{\mathbb{R}}^{d}}\{e_{1},\cdots,e_{d}\}$ to the geodesic simplex $S$. According to Lemma 2.2 (ii), $\Phi$ maps convex sets to geodesic convex sets, and by (3.1) and Lemma 2.2 (iv),

[TABLE]

Since $\Phi$ maps boundary of $T_{0}$ to boundary of $S$, (3.2) in particular implies that if $B(x,t)_{H_{T_{0}}}\subset T_{0}$ for some $x\in T_{0}$ and $t\in(0,1)$, then

[TABLE]

On the other hand, using Lemma 2.2 (iii), we obtain by a change of variable that for each nonnegative function $f$ on $S$,

[TABLE]

Thus, setting $w_{0}(x)=a|\text{det}A|w(\Phi(x))/\|Ax\|^{d}$ for $x\in T_{0}$, we have $Nw_{0}(T_{0})=Nw(S)\in{\mathbb{N}}$ and

[TABLE]

with $c=1/c_{1,d}\varepsilon^{2}$. Thus, according to Proposition 2.6, $T_{0}$ has a convex partition $T_{0}=\bigcup_{j=1}^{m}T_{0,j}$ with the properties that $B(x_{j},c_{d}r)_{H_{T_{0}}}\subset T_{0,j}\subset B(x_{j},c_{d}^{\prime}r)_{H_{T_{0}}}$ and $Nw_{0}(T_{0,j})\in{\mathbb{N}}$ for all $1\leq j\leq m$. Setting

[TABLE]

and using (3.3) and Lemma 2.2, we conclude that $\{R_{1},\cdots,R_{m}\}$ is a convex partition of the geodesic simplex $S$ with the stated properties in Proposition 3.2.

∎

Finally, we turn to the proof of Proposition  Propsition 3.3, which is more involved. The important ingredient used in the proof is a family of nonlinear dilations on $\mathbb{S}^{d-1}$, which we shall introduce and study in the subsection that follows.

3.1. A family of nonlinear dilations

Throughout this subsection, we write $\mathbf{x}(\varphi,\xi)=(\cos\varphi,\xi\sin\varphi)\in\mathbb{S}^{d-1}$ for $(\varphi,\xi)\in[0,\pi]\times\SS^{d-2}$.

Definition 3.4**.**

For $0\leq{\alpha}<{\beta}\leq 2\pi$, define

[TABLE]

where it is agreed that $\SS^{0}=\{\pm 1\}$ when $d=3$.

Next, for $0<{\alpha}<\frac{\pi}{2}$, we define the function $h_{\alpha}:[-1,1]\to(0,1]$ by

[TABLE]

It is easily seen that $\frac{2}{\pi}{\alpha}\leq h_{\alpha}(t)\leq 1$ for $t\in[-1,1]$ and

[TABLE]

Using the function $h_{\alpha}$, we may define a nonlinear dilation $T_{\alpha}x$ of $x\in\mathbb{S}^{d-1}$ with respect to the angle between $x$ and $e_{1}$ as follows:

Definition 3.5**.**

Given $0<{\alpha}<\frac{\pi}{2}$, define the mapping $T_{\alpha}:\SS^{d-1}\setminus\{-e_{1}\}\to\SS^{d-1}\setminus\{-e_{1}\}$ by T_{\alpha}(x):=\mathbf{x}\bigl{(}h_{\alpha}(\xi_{1})\varphi,\xi\bigr{)} for $x=\mathbf{x}(\varphi,\xi)\in\SS^{d-1}\setminus\{-e_{1}\}$ with $\varphi\in[0,\pi)$ and $\xi=(\xi_{1},\cdots,\xi_{d-1})\in\SS^{d-2}$.

The following lemma collects some useful properties of $T_{\alpha}$:

Lemma 3.6**.**

Let $0<{\alpha}<\frac{\pi}{2}$ and let $\SS_{0}^{d-1}:=\{x\in\SS^{d-1}:\ \ x_{1}=0,\ x_{2}\geq 0\}$. Then the following statements hold:

1. (i)

If $x,y\in\mathbb{S}^{d-1}$ and $0\leq\mathtt{d}(x,e_{1}),\mathtt{d}(y,e_{1})\leq\frac{3\pi}{4}$, then

[TABLE] 2. (ii)

$T_{\alpha}$* is a bijective mapping from $S(0,\frac{\pi}{2})$ to $S(0,{\alpha})$ that maps boundary of $S(0,\frac{\pi}{2})$ to the boundary of $S(0,{\alpha})$.* 3. (iii)

For every nonnegative measurable function $f$ on $S(0,{\alpha})$,

[TABLE]

where

[TABLE] 4. (iv)

If $S\subset S(0,\frac{\pi}{2})$ is geodesic simplex spanned by $e_{1}$ and a set of independent vectors $\eta_{1},\cdots,\eta_{d-1}\in\SS_{0}^{d-1}$, then the set $T_{\alpha}(S)\subset S(0,{\alpha})$ is a geodesic simplex spanned by the set $\{T_{\alpha}(\eta_{1}),T_{\alpha}(\eta_{2}),\ldots,T_{\alpha}(\eta_{d-1}),e_{1}\}$. 5. (v)

If $\{\eta_{1},\cdots,\eta_{d-1}\}$ is a set of strongly $\varepsilon$-separated points in $\SS_{0}^{d-1}$ for some $\varepsilon\in(0,1)$, then $\{T_{\alpha}(\eta_{1}),\cdots,T_{\alpha}(\eta_{d-1}),e_{1}\}$ is a set of strongly $c_{d,{\alpha}}\varepsilon$-separated points on $\mathbb{S}^{d-1}$.

Proof.

(i) Assume that $x=\mathbf{x}({\theta}_{1},\xi)$ and $y=\mathbf{x}({\theta}_{2},\eta)$ with $0\leq{\theta}_{1}\leq{\theta}_{2}\leq\frac{3\pi}{4}$. According to Lemma 2.3, we have

[TABLE]

and

[TABLE]

By (3.4),

[TABLE]

which combined with (3.7) and (3.8) implies that $\mathtt{d}(T_{\alpha}x,T_{\alpha}y)\leq C_{d,{\alpha}}\mathtt{d}(x,y)$. On the other hand, a similar argument shows that

[TABLE]

which, using (3.7) and (3.8) once again, implies the inverse inequality $\mathtt{d}(x,y)\leq\mathtt{d}(T_{\alpha}x,T_{\alpha}y)$. This completes the proof of (i).

(ii) Since $(\varphi,\xi)\mapsto\mathbf{x}(\varphi,\xi)$ is an injective mapping from $(0,\pi)\times\SS^{d-2}$ to $\mathbb{S}^{d-1}$, it follows that $T_{\alpha}:\mathbb{S}^{d-1}\setminus\{-e_{1}\}\to\mathbb{S}^{d-1}\setminus\{-e_{1}\}$ is injective. To show that $T_{\alpha}$ is a mapping from $S(0,\frac{\pi}{2})$ onto $S(0,{\alpha})$, we need the following fact, which is a direct consequence of Definition 3.4:

Fact 1.    If $0<{\alpha}\leq\frac{\pi}{2}$, then $\mathbf{x}(\varphi,\xi)\in S(0,{\alpha})$ if and only if $\varphi=0$ or $0<\varphi\leq\frac{\pi}{2}$, $\xi_{1}\geq 0$ and $\tan\varphi\leq\frac{\tan{\alpha}}{\xi_{1}}$, where is agreed that $\tan\frac{\pi}{2}=\infty$ and $\frac{\tan{\alpha}}{0}=\infty$.

Note also that according to Definition 3.5, if $x=(x_{1},x_{2},\cdots,x_{d})\in\mathbb{S}^{d-1}\setminus\{-e_{1}\}$ with $x_{2}=0$, then $T_{\alpha}x=x$, which in particular implies $T_{\alpha}(e_{1})=e_{1}\in S(0,{\alpha})$. If $x=\mathbf{x}(\varphi,\xi)\in S(0,\pi/2)\setminus\{e_{1}\}$, then $0<\varphi\leq\frac{\pi}{2}$, $\xi_{1}\geq 0$ and $\tan(h_{\alpha}(\xi_{1})\varphi)\leq\tan(\frac{\pi}{2}h_{\alpha}(\xi_{1}))=\frac{\tan{\alpha}}{\xi_{1}},$ which, by Fact 1, implies that $T_{\alpha}(x)\in S(0,{\alpha})$. Conversely, if $y=\mathbf{x}(\varphi_{1},\xi)\in S(0,{\alpha})\setminus\{e_{1}\}$, then by Fact 1, $0<\varphi_{1}\leq\frac{\pi}{2}$, $0\leq\xi_{1}\leq 1$, $\tan(\varphi_{1})\leq\frac{\tan{\alpha}}{\xi_{1}}$, and hence

[TABLE]

This implies that $x=\mathbf{x}(\frac{\varphi_{1}}{h_{\alpha}(\xi_{1})},\xi)\in S(0,\frac{\pi}{2})$ and $y=T_{\alpha}(x)$. Finally, we show $T_{\alpha}$ maps boundary of $S(0,\frac{\pi}{2})$ to boundary of $S(0,{\alpha})$. Indeed, if $x=(r,0\sqrt{1-r^{2}}\eta)\in S(0,\frac{\pi}{2})$ with $0\leq r\leq 1$, then $T_{\alpha}x=x\in S(0,{\alpha})$. On the other hand, it can easily verified that if $x=(0,\xi)\in S(0,\frac{\pi}{2})$ with $\xi_{1}>0$, then $T_{\alpha}x=(r\cos{\alpha},\frac{(r\sin{\alpha})\xi}{\xi_{1}})\in S(0,{\alpha})$ with $r=\frac{\xi_{1}}{\sqrt{\xi_{1}^{2}\cos^{2}{\alpha}+\sin^{2}{\alpha}}}$.

(iii) Using Fact 1, we obtain

[TABLE]

which, by a change of variable $\varphi=h_{\alpha}(\xi_{1}){\theta}$, equals

[TABLE]

(iv) We start with the proof of the following fact:

Fact 2.    If $x=(0,\bar{x})\in\mathbb{S}^{d-1}_{0}$, then $T_{\alpha}x=\frac{E_{\alpha}(x)}{\|E_{\alpha}(x)\|},$ where $E_{\alpha}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ is a linear operator given by $E_{\alpha}y=(y_{2}\cos{\alpha},\bar{y}\sin{\alpha})$ for $y=(y_{1},y_{2},\cdots,y_{d})=(y_{1},\bar{y})\in{\mathbb{R}}^{d}$.

To see this, let $x=(0,x_{2},\cdots,x_{d})\in\mathbb{S}^{d-1}_{0}$. If $x_{2}=0$, then $T_{\alpha}x=x$ and Fact 2 holds trivially. If $x_{2}>0$, then setting $h=\frac{\pi}{2}h_{\alpha}(x_{2})=\tan^{-1}\frac{\tan{\alpha}}{x_{2}}$, we have

[TABLE]

This proves Fact 2.

Next, we prove that for $V=\text{conv}_{\mathbb{S}^{d-1}}\{\eta_{1},\cdots,\eta_{d-1}\},$

[TABLE]

Indeed, if $x\in V$, then $x=\sum_{j=1}^{d-1}t_{j}\eta_{j}\in\SS_{0}^{d-1}$ for some $t_{1},\cdots,t_{d-1}\geq 0$, and hence, by Fact 2,

[TABLE]

which implies that $T_{\alpha}(x)\in\text{conv}_{\mathbb{S}^{d-1}}\{T_{\alpha}(\eta_{1}),\ldots,T_{\alpha}(\eta_{d-1})\}$. Conversely, if $y\in\text{conv}_{\mathbb{S}^{d-1}}\{T_{\alpha}(\eta_{1}),\ldots,T_{\alpha}(\eta_{d-1})\}$, then $y=\sum_{j=1}^{d-1}s_{j}T_{\alpha}(\eta_{j})$ for some $s_{1},\cdots,s_{d-1}\geq 0$, and by Fact 2,

[TABLE]

where $s_{j}^{\prime}=\frac{s_{j}}{\|E_{\alpha}(\eta_{j})\|}$ and $c_{1}=\|\sum_{j=1}^{d-1}s_{j}^{\prime}\eta_{j}\|$.

Third, we show that for any $\eta\in\SS_{0}^{d-1}$,

[TABLE]

Assume that $\eta=(0,\xi)$ with $\xi=(\xi_{1},\cdots,\xi_{d-1})\in\SS^{d-2}$ and $\xi_{1}\geq 0$. Then \operatorname{Arc}(e_{1},\eta)=\Bigl{\{}(\cos{\theta},\xi\sin{\theta}):\ \ {\theta}\in[0,\frac{\pi}{2}]\Bigr{\}}. If $x=(\cos\varphi,\xi\sin\varphi)\in\operatorname{Arc}(e_{1},\eta)$ with $\varphi\in[0,\frac{\pi}{2}]$, then setting $h_{\alpha}=\frac{2}{\pi}\tan^{-1}\frac{\tan{\alpha}}{\xi_{1}}$, we have

[TABLE]

where $s_{1}=\frac{\sin(h_{\alpha}\varphi)}{\sin{\alpha}}\|E_{\alpha}(\eta)\|\geq 0$ and $s_{2}=\cos(h_{\alpha}\varphi)-\frac{\sin(h_{\alpha}\varphi)}{\tan{\alpha}}\xi_{1}$. Since $\tan(h_{\alpha}\varphi)\leq\tan(\frac{\pi}{2}h_{\alpha})=\frac{\tan{\alpha}}{\xi_{1}},$ it follows that $s_{2}\geq 0$, and hence $T_{\alpha}x\in\operatorname{Arc}(e_{1},T_{\alpha}(\eta))$. Conversely, if $y=t_{1}e_{1}+t_{2}T_{\alpha}(\eta)\in\operatorname{Arc}(e_{1},T_{\alpha}(\eta))$ for some $t_{1},t_{2}\geq 0$, then

[TABLE]

Finally, we prove

[TABLE]

Indeed, it is easily seen that each $x\in S$ can be written in the form $x=\sqrt{1-t^{2}}\eta+te_{1}\in\operatorname{Arc}(e_{1},\eta)$ for some $t\geq 0$ and $\eta\in V=\text{conv}_{\mathbb{S}^{d-1}}\{\eta_{1},\cdots,\eta_{d-1}\}$. It follows by (3.10) that $T_{\alpha}(x)\in\operatorname{Arc}(e_{1},T_{\alpha}(\eta))$. However, by (3.9),

[TABLE]

Thus,

[TABLE]

Conversely, if $y\in\text{conv}_{\mathbb{S}^{d-1}}\{T_{\alpha}(\eta_{1}),\cdots,T_{\alpha}(\eta_{d-1}),e_{1}\}$, then $y=s_{0}e_{1}+\sum_{j=1}^{d-1}s_{j}T_{\alpha}(\eta_{j})$ for some $s_{0},s_{1},\cdots,s_{d-1}\geq 0$. According to (3.9), there exists $\eta\in V$ such that $\sum_{j=1}^{d-1}s_{j}T_{\alpha}(\eta_{j})=c_{2}T_{\alpha}(\eta),$ with $c_{2}=\|\sum_{j=1}^{d-1}s_{j}T_{\alpha}(\eta_{j})\|$. It then follows by (3.10) that

[TABLE]

(v)   According to Fact 2, we have

[TABLE]

By symmetry, it remains to show that

[TABLE]

Indeed, since $\sin{\alpha}\leq\|E_{\alpha}(\eta_{1})\|\leq\sqrt{2}$, we obtain from Fact 2 that

[TABLE]

∎

3.2. Proof of Proposition 3.3

Without loss of generality, we may assume that $N\geq N_{d,w}$. Given $0\leq{\alpha}<{\beta}\leq 2\pi$, let $S({\alpha},{\beta})$ denote the subset of $\mathbb{S}^{d-1}$ given in Definition 3.4. The following assertion plays an important role in the proof of Proposition 3.3.

Assertion A.    If $Nw(S(0,\frac{\pi}{2}))\in{\mathbb{N}}$, then there exists a convex partition $\{S_{1},\cdots,S_{n_{0}}\}$ of the set $S(0,\frac{\pi}{2})$ such that each $S_{j}$ is a geodesic simplex spanned by the vector $e_{1}$ and a set of $d-1$ strongly $\varepsilon_{d}$-separated points in $\SS_{0}^{d-1}=\{x\in\mathbb{S}^{d-1}:\ \ x_{1}=0,\ \ x_{2}\geq 0\}$, and satisfying the condition $Nw(S_{j})\in{\mathbb{N}}$.

For the moment, we take Assertion A for granted and proceed with the proof of the proposition. Note that if $0\leq{\alpha}<{\beta}\leq 2\pi$ and ${\beta}-{\alpha}\geq\frac{\pi}{6}$, then $S({\alpha},{\beta})$ contains a spherical cap of radius $\geq\frac{1}{20}$. It follows by continuity that $\mathbb{S}^{d-1}$ has a partition $\mathbb{S}^{d-1}=\bigcup_{j=1}^{\ell_{0}}S({\alpha}_{j-1},{\alpha}_{j})$ with $1\leq\ell_{0}\leq 12$ such that ${\alpha}_{0}=0$, $\frac{\pi}{6}\leq{\alpha}_{i}-{\alpha}_{i-1}<\frac{\pi}{2}$ and $Nw(S({\alpha}_{i-1},{\alpha}_{i}))\in{\mathbb{N}}$ for $i=1,\cdots,\ell_{0}$. Thus, replacing $w$ with $w\circ Q$ for some rotations $Q\in SO(d)$, we reduce to proving the following assertion:

Assertion B:   If ${\alpha}\in[\frac{\pi}{6},\frac{\pi}{2}]$ and $Nw(S(0,{\alpha}))\in{\mathbb{N}}$, then there exists a convex partition $\{S_{1},\cdots,S_{n_{0}}\}$ of the set $S(0,{\alpha})$ such that $S_{j}\in\mathcal{S}_{\varepsilon_{d}}$ and $Nw(S_{j})\in{\mathbb{N}}$ for $j=1,\cdots,n_{0}$.

The proof of Assertion B relies on Assertion A and the nonlinear bijective mapping $T_{\alpha}:S(0,\frac{\pi}{2})\to S(0,{\alpha})$ introduced in Definition 3.5. Let $w_{1}(x):=w(T_{\alpha}(x))\omega(x)$ for $x\in S(0,\frac{\pi}{2})$, where $\omega$ is the function given in (3.6). According to (3.5), $w_{1}(E)=w(T_{\alpha}(E))$ for each $E\subset S(0,\frac{\pi}{2})$. Moreover, by By Lemma 3.6 (i), (ii) and (iii), we may apply Assertion A to the weight $w_{1}$ on $S(0,\frac{\pi}{2})$ and obtain a partition $\{V_{1},\cdots,V_{n_{0}}\}$ of $S(0,\frac{\pi}{2})$ with the stated properties of Assertion A (with $w_{1}$ in place of $w$). Now setting $S_{j}=T_{\alpha}(V_{j})$, we get a partition $\{S_{1},\cdots,S_{n_{0}}\}$ of $S(0,{\alpha})$. By Lemma 3.6 (iv) and (v), each $S_{j}$ is a geodesic simplex in the class $\mathcal{S}_{\varepsilon_{d}}$, whereas by Lemma 3.6 (iii), $Nw(S_{j})=Nw_{1}(T_{j})\in{\mathbb{N}}$ for $1\leq j\leq n_{0}$. This proves Assertion B.

It remains to show Assertion A. We start with the case of $d=3$. Note that

[TABLE]

By continuity, there exists a point $\xi=(0,\sin{\theta},\cos{\theta})\in\SS_{0}^{2}$ for some ${\theta}\in(\frac{\pi}{6},\frac{\pi}{2})$ such that $S(0,\frac{\pi}{2})=S_{1}\cup S_{2}$ with $S_{1}:=\text{conv}_{\SS^{2}}\{e_{1},-e_{3},\xi\}$ and $S_{2}:=\text{conv}_{\SS^{2}}\{e_{1},e_{3},\xi\}$, and such that $Nw(S_{1}),Nw(S_{2})\in{\mathbb{N}}$. This shows Assertion A and hence Proposition 3.3 for $d=3$.

Next, we show Assertion A for $d\geq 4$. We use induction on the dimension $d$. Assume that the conclusion of Proposition 3.3 holds on the spheres $\SS^{\ell-1}$, $\ell=3,\cdots,d-1$. We shall prove that Assertion A holds for $d\geq 4$, which in turn implies Proposition 3.3 for $d\geq 4$. The proof relies on the following formula:

[TABLE]

where $f$ is a nonnegative function on $S(0,\pi/2)$.

For $\eta\in\SS^{d-3}$, define

[TABLE]

where $\omega=w(S(0,\pi/2))=\frac{k_{0}}{N}$ for some positive integer $k_{0}$. Given a set $E\subset\SS^{d-3}$, set

[TABLE]

It can be easily seen that if $E$ is a geodesic simplex in $\SS^{d-3}$ spanned by a set of linearly independent points $\eta_{1},\cdots,\eta_{d-2}\in\SS^{d-3}$, then $\widehat{E}$ is a geodesic simplex in $\SS^{d-1}$ spanned by the set $\{e_{1},e_{2},\widehat{\eta}_{1},\cdots,\widehat{\eta}_{d-2}\}$, with $\widehat{\eta}_{j}=(0,0,\eta_{j})\in\SS_{0}^{d-1}$ for $j=1,\cdots,d-2$, and moreover, by (3.13), $\widetilde{w}(E)=\omega^{-1}w(\widehat{E})$. Now applying the induction hypothesis to the weight $\widetilde{w}$ on the sphere $\SS^{d-3}$ with $k_{0}$ in place of $N$, we obtain a partition $\{E_{1},\cdots,E_{m_{0}}\}$ of the sphere $\SS^{d-3}$, where each $E_{j}$ is a geodesic simplex in $\SS^{d-3}$ spanned by a set of strongly $\varepsilon_{d-2}$-separated points $\eta_{j,1},\cdots,\eta_{j,d-2}\in\SS^{d-3}$ and satisfying $k_{0}\widetilde{w}(E_{j})\in{\mathbb{N}}$. It then follows that $\{\widehat{E}_{1},\cdots,\widehat{E}_{m_{0}}\}$ is a partition of $S(0,\frac{\pi}{2})$ where each $\widehat{E}_{j}$ is a geodesic simplex spanned by the vector $e_{1}$ and the set of strongly $\varepsilon_{d}$-separated points $\widehat{\eta}_{j,1},\cdots,\widehat{\eta}_{j,d-2},e_{2}\in\SS_{0}^{d-1}$, and satisfies that $Nw(\widehat{E}_{j})=k_{0}\widetilde{w}(E_{j})\in{\mathbb{N}}$ for each $j$. This proves Assertion A for $d\geq 4$.

4. Preliminary lemmas

For the proof of Theorem 1.1, in addition to the convex partitions of the weighted sphere (i.e., Theorem 1.4), we shall also need several preliminary lemmas, which we state or prove in this section.

Throughout this section, $w$ denotes a normalized doubling weight on $\mathbb{S}^{d-1}$ with doubling constant $L_{w}$. All the general constants $c_{w},C_{w},{\delta}_{w}$ depend only on $d$ and the doubling constant of $w$. Given $1\leq p\leq\infty$, we denote by $\|\cdot\|_{p,w}$ the $L^{p}$-norm defined with respect to the measure $w(x)d{\sigma}(x)$ on $\mathbb{S}^{d-1}$. A finite subset $\Lambda$ of $\mathbb{S}^{d-1}$ is called $\varepsilon$-separated for a given $\varepsilon>0$ if $\mathtt{d}({\omega},{\omega}^{\prime})\geq\varepsilon$ for every two distinct points ${\omega},{\omega}^{\prime}\in\Lambda$. A $\varepsilon$-separated subset $\Lambda$ is called maximal if $\bigcup_{{\omega}\in\Lambda}B({\omega},\varepsilon)=\mathbb{S}^{d-1}$.

The following weighted polynomial inequalities were established in [9] (see also [10, Theorem 5.3.6.] and [10, Theorem 5.3.4.]):

Lemma 4.1**.**

[9]** Let $n$ be a positive integer and ${\delta}$ a parameter in $(0,1)$. Assume that $1\leq p<\infty$ and $\Lambda$ is a $\frac{{\delta}}{n}$-separated subset of $\mathbb{S}^{d-1}$ .

1. (i)

For every spherical polynomial $f\in\Pi_{n}^{d}$,

[TABLE]

where $B_{\omega}=B(\omega,n^{-1}{\delta})$ and $\operatorname{osc}(f)(\omega)=\max_{y\in B_{\omega}}|f(y)|$ for $\omega\in\Lambda$. 2. (ii)

If, in addition, $\Lambda$ is maximal $\frac{{\delta}}{n}$-separated with $0<{\delta}<c_{w}$ and $c_{w}$ being a small constant, then for each given parameter ${\gamma}>1$, and every $f\in\Pi_{n}^{d}$,

[TABLE]

where $B_{\omega}=B(\omega,n^{-1}{\delta})$ for $\omega\in\Lambda$, and the constants of equivalence depend only on $d$, ${\gamma}$ and the doubling constant of $w$.

The weighted Christoffel function on $\mathbb{S}^{d-1}$ is defined by

[TABLE]

where the infimum is taken over all spherical polynomials of degree $n$ on $\mathbb{S}^{d-1}$ that take the value $1$ at the point $x\in\mathbb{S}^{d-1}$. The following pointwise estimate of ${\lambda}_{n}(w,x)$ is proved in [11]. (See also [31] for the case of $d=2$).

Lemma 4.2**.**

[11]** For $x\in\mathbb{S}^{d-1}$ and $n\in{\mathbb{N}}$,

[TABLE]

where the constant of equivalence depends only on $d$ and $L_{w}$.

The tangential gradient of $f\in C^{1}({\mathbb{R}}^{d})$ is defined as

[TABLE]

where F(y)=F_{x}(y)=f\Bigl{(}\frac{\|x\|y}{\|y\|}\Bigr{)} for $y\in{\mathbb{R}}^{d}\setminus\{0\}$.

We shall need the following lemma from [5], whose detailed proof can be found in the book [10, p.145-147].

Lemma 4.3**.**

[5]** For $\varepsilon>0$, $x\in\mathbb{S}^{d-1}$ and each fixed spherical polynomial $P\in\Pi_{n}^{d}$, the differential equation

[TABLE]

has a unique solution $y=y(P,s)\equiv y(P,x;s)\in\mathbb{S}^{d-1}$ on $[0,\infty)$ with the following properties:

1. (i)

$y(P,x;s)\in\mathbb{S}^{d-1}$* for all $s\geq 0$, $x\in\mathbb{S}^{d-1}$ and $P\in\Pi_{n}^{d}$;* 2. (ii)

for each fixed $s\geq 0$ and $x\in\mathbb{S}^{d-1}$, $P\mapsto y(P,x;s)$ is a continuous mapping from $\Pi_{n}^{d}$ to $\mathbb{S}^{d-1}$; 3. (iii)

If $x,x^{\prime}\in\mathbb{S}^{d-1}$ and $x\neq x^{\prime}$, then

[TABLE]

As a consequence of Lemma 4.3, we have the following useful corollary:

Corollary 4.4**.**

Let $R\subset\mathbb{S}^{d-1}$ be a closed geodesically convex subset of $\mathbb{S}^{d-1}$ with $\max_{x,y\in R}\mathtt{d}(x,y)<\frac{\pi}{2}$. Assume that $P\in\Pi_{n}^{d}$ and $z_{\max}\in R$ is such that $P(z_{\max})=\max_{z\in R}P(z)$. Then for each $x\in R$,

[TABLE]

where $\mathtt{d}(x,\partial(R))=\inf_{y\in\partial(R)}\mathtt{d}(x,y)$.

Corollary 4.4 was used without proof in [6]. For completeness, we include a proof here.

Proof.

Without loss of generality, we may assume that $x$ is an interior point of $R$ and $\min_{z\in R}|\nabla_{0}P(z)|=\varepsilon>0$. By Lemma 4.3, there exists a continuously differentiable function $y:[0,\infty)\to\mathbb{S}^{d-1}$ satisfying the equation (4.3). We claim that there exists $s>0$ such that $y(s)\notin R$. Assuming otherwise, we then have that

[TABLE]

and hence, for any $t>0$,

[TABLE]

Letting $t\to\infty$, and taking into account the fact that $y(t)\in\mathbb{S}^{d-1}$, we conclude that the polynomial $P$ is not bounded on $\mathbb{S}^{d-1}$, which contradicts the extreme value theorem. This proves the claim.

Now set

[TABLE]

Since $y:[0,\infty)\to\mathbb{S}^{d-1}$ is continuous and $x$ is an interior point of $R$, it follows that $0<t_{0}<\infty$, $y(t_{0})\in\partial(R)\subset R$, and

[TABLE]

This implies that

[TABLE]

∎

We will use the following lemma from algebraic topology.

Lemma 4.5**.**

[33, Ths. 1.2.6 and 1.2.9]** Let $V$ be a finite dimensional real Hilbert space with inner product ${\langle}\cdot,\cdot{\rangle}$, and $\Omega$ a bounded open subset of $V$ which contains the zero vector in $V$. If $f:V\to V$ is a continuous mapping on the space $V$ with the property ${\langle}f(x),x{\rangle}>0$ for all $x\in\partial\Omega$, then there exists a vector $x_{0}\in\Omega$ such that $f(x_{0})=0$.

For convenience, we introduce the following definition:

Definition 4.6**.**

For $x,y\in\mathbb{S}^{d-1}$ with ${\theta}:=\mathtt{d}(x,y)\in(0,\frac{\pi}{2})$, let ${\gamma}_{[x,y]}:[0,1]\to\operatorname{Arc}(x,y)$ denote the parametric representation of $\operatorname{Arc}(x,y)$given by

[TABLE]

where $\xi\in\mathbb{S}^{d-1}$ is such that $\xi\sin{\theta}$ is the orthogonal projection of $y$ on the space $\{y\in{\mathbb{R}}^{d}:\ \ y\cdot x=0\}$; that is, $\xi=\xi_{x,y}=\frac{y-x\cos{\theta}}{\sin{\theta}}$ . In the case when $x=y$, we also set ${\gamma}_{[x,y]}(t)=x$ for $t\in[0,1]$.

A convex set $G$ in ${\mathbb{R}}^{d}$ is said to be strictly convex if for any two distinct points $p,q\in G$ and any $t\in(0,1)$, $tp+(1-t)q$ is an interior point of $G$.

Lemma 4.7**.**

[6]** Let $R$ be a closed geodesically convex subset of $\mathbb{S}^{d-1}$ with $\max_{p,q\in R}\mathtt{d}(p,q)<\pi/2$. Assume that $x_{0}$ is a given interior point of $R$ and let $T_{0}:=\{y\in{\mathbb{R}}^{d}:\ \ x_{0}\cdot y=0\}$. Then the following statements hold:

1. (i)

For each $y\in T_{0}\setminus\{0\}$, there exists a unique point $x_{y}\in R$ such that $x_{y}\cdot y=\max_{z\in R}z\cdot y$. Furthermore, $y\longmapsto x_{y}$ is a continuous mapping on the set $T_{0}\setminus\{0\}$ with the property that the function $t\mapsto y\cdot({\gamma}_{[x_{y},w]}(t))$ is decreasing on $[0,1]$ for each given $y\in T_{0}\setminus\{0\}$ and $w\in R\setminus\{x_{y}\}$. 2. (ii)

Let $z_{0}$ be an arbitrary interior point of $R$. Given ${\delta},\varepsilon\in(0,1/2)$, define

[TABLE]

Then $y\mapsto A(y)$ is a continuous mapping from $T_{0}$ to $R$.

Lemma 4.7 was essentially proved in [6]. However, since the proof there is rather sketchy, we include a more detailed proof of the lemma here.

Proof.

(i) Let $P_{0}(z)=z-(z\cdot x_{0})x_{0}$ denote the orthogonal projection of $z\in\mathbb{S}^{d-1}$ onto the space $T_{0}$. Firstly, we prove that $D_{0}=P_{0}(R)$ is a strictly convex set in the space $T_{0}$. To see this, let $S_{0}^{+}=\{y\in\mathbb{S}^{d-1}:\ \ y\cdot x_{0}>0\}$ and $U_{0}:=\{y\in T_{0}:\ \ \|y\|<1\}$. Clearly, $R\subset S_{0}^{+}$, $D_{0}\subset U_{0}$, and $P_{0}$ is a continuous mapping from $S_{0}^{+}$ onto the set $U_{0}$ with continuous inverse given by $P^{-1}(u)=u+\sqrt{1-\|u\|^{2}}x_{0}$ for $u\in U_{0}$. Thus, to show $D_{0}$ is strictly convex, it suffices to prove that for any two distinct points $u,v\in D_{0}$, and every $w=tu+(1-t)v$ with $t\in(0,1)$, $P_{0}^{-1}(w)$ is an interior point of $R$. Note first that

[TABLE]

with ${\alpha}_{0}=\sqrt{1-\|w\|^{2}}-t\sqrt{1-\|u\|^{2}}-(1-t)\sqrt{1-\|v\|^{2}}$. We claim that ${\alpha}_{0}>0$. Indeed, since $g(x)=\|x\|^{2}$ is a strictly convex function on ${\mathbb{R}}^{d}$, we have $\|w\|^{2}<t\|u\|^{2}+(1-t)\|v\|^{2}$. Since $\varphi(s)=\sqrt{s}$ is a concave function on $[0,\infty)$, this implies that

[TABLE]

which in turn implies that ${\alpha}_{0}>0$. Now setting

[TABLE]

we have $p\in\operatorname{Arc}(P_{0}^{-1}(u),P_{0}^{-1}(v))\subset R$. (4.6) then implies that $P_{0}^{-1}(w)\in\operatorname{Arc}(p,x_{0})\subset R$. To show that $P_{0}^{-1}(w)$ is in fact an interior point of $R$, let ${\delta}\in(0,1)$ be such that $\{\frac{x_{0}+\eta}{\|x_{0}+\eta\|}:\ \ \|\eta\|<{\delta}\}\subset R$. Then for $z=P_{0}^{-1}(w)+{\alpha}_{0}\eta\in\mathbb{S}^{d-1}$ with $\eta\in{\mathbb{R}}^{d}$ satisfying $\|\eta\|<{\delta}$, we use (4.6) to obtain

[TABLE]

This shows that $P_{0}^{-1}(w)$ is an interior point of $R$, and hence proves that $D_{0}$ is a strictly convex subset of $T_{0}$.

Secondly, we show that for each $y\in T_{0}\setminus\{0\}$, there exists a unique $x_{y}\in R$ such that $x_{y}\cdot y=\max_{z\in R}z\cdot y$. Indeed, this follows directly from the facts that $D_{0}$ is strictly convex and $\max_{z\in R}z\cdot y=\max_{z\in R}P_{0}(z)\cdot y$.

Thirdly, we show that $y\mapsto x_{y}$ is continuous on $T_{0}\setminus\{0\}$. Let $y,z$ be two distinct nonzero vectors in $T_{0}$, and let $p=P_{0}(x_{y})$ and $q=P_{0}(x_{z})$. Then $p,q\in D_{0}$, $p\cdot y=x_{y}\cdot y=\max_{u\in D_{0}}u\cdot y$, and $q\cdot z=x_{z}\cdot z=\max_{u\in D_{0}}u\cdot z$. Since $\max_{x,y\in R}\mathtt{d}(x,y)<\frac{\pi}{2}$, we also have that $\|x_{y}-x_{z}\|\leq C_{R}\|p-q\|$. For convenience, we set $H_{p,y}:=\{u\in T_{0}:\ \ u\cdot y=p\cdot y\}$, $H_{p,y}^{-}:=\{u\in T_{0}:\ \ u\cdot y\leq p\cdot y\}$ and $H_{p,z}^{+}:=\{u\in T_{0}:\ \ u\cdot z\geq p\cdot z\}$. Clearly, $q\in H_{p,z}^{+}\cap D_{0}$ and $D_{0}\subset H_{p,y}^{-}$. Note also that

[TABLE]

Since $p\cdot z\leq q\cdot z$, it follows that

[TABLE]

On the other hand, since $D_{0}$ is strictly convex, $\rho(\frac{p+u}{2},H_{p,y})>0$ for any $u\in D_{0}\setminus\{p\}$. Thus, given any $\varepsilon>0$,

[TABLE]

In particular, if $\|x_{y}-x_{z}\|\geq\varepsilon/C_{R}$, then $\|p-q\|\geq\varepsilon$ and hence by (4.7), $0<{\delta}\|y\|\leq\|y-z\|$. This shows the continuity of the mapping $y\mapsto x_{y}$.

Finally, we show that given each $y\in T_{0}\setminus\{0\}$ and $w\in R\setminus\{x_{y}\}$, the function $t\mapsto y\cdot({\gamma}_{[x_{y},w]}(t))$ is decreasing on $[0,1]$. Let $G$ denote the great circle passing through $x_{y}$ and $w$, and $\eta$ the orthogonal projection of $y$ onto the plane spanned by the vectors $x_{y}$ and $w$. Let $\xi=\frac{\eta}{\|\eta\|}\in G$. Then

[TABLE]

Since $w\cdot\xi\leq x_{y}\cdot\xi$, it suffices to show that the arc $\operatorname{Arc}(x_{y},w)$ lies between the points $\xi$ and $-\xi$ on the great circle $G$. Indeed, since $\max_{u,v\in R}\mathtt{d}(u,v)<\frac{\pi}{2}$, we have $x_{y}\cdot\xi=\|\eta\|^{-1}x_{y}\cdot y>0$. Hence, $\xi$ cannot be in the interior of the geodesic arc $\operatorname{Arc}(w,x_{y})$ since otherwise $\xi\in R$ and $\|\eta\|=\xi\cdot\eta=\xi\cdot y>x_{y}\cdot y$. Similarly, one can also show that that $-\xi$ can not lie in the interior of $\operatorname{Arc}(w,x_{y})$. Indeed, assuming otherwise, we have that $-\xi\in R$, which would imply that $x_{y}\cdot y=\|\eta\|\xi\cdot x_{y}<0$, yielding a contradiction.

(ii) Let ${\delta}_{1}\in(0,\frac{1}{2})$ be such that $B(z_{0},{\delta}_{1})\subset R$. Since $x_{y}$ is on the boundary of $R$, it follows by (i) that $\theta(y)=\mathtt{d}(x_{y},z_{0})\in[{\delta}_{1},\frac{\pi}{2})$ for each $y\in T_{0}\setminus\{0\}$, and is continuous in $y\in T_{0}\setminus\{0\}$. Furthermore, according to (4.4), for $y\in T_{0}\setminus\{0\}$,

[TABLE]

where

[TABLE]

Since $0<{\delta}_{1}\leq{\theta}(y)<\frac{\pi}{2}$, $y\mapsto\xi_{y}$ is a continuous mapping from $T_{0}\setminus\{0\}$ to $\mathbb{S}^{d-1}$. Since the function $h_{\varepsilon}(y):=\min\{1,\frac{\|y\|}{\varepsilon}\}$ is continuous on $T_{0}$, it follows by (4.8) that A(y)={\gamma}_{[z_{0},x_{y}]}\Bigl{(}(1-{\delta})h_{\varepsilon}(y)\Bigr{)} is continuous on $T_{0}\setminus\{0\}$. On the other hand, setting $t_{y}:=(1-{\delta})h_{\varepsilon}(y)\theta(y)$ for $y\in T_{0}\setminus\{0\}$, we have that $\lim_{\begin{subarray}{c}y\to 0\\ y\in T_{0}\end{subarray}}t_{y}=0,$ and hence

[TABLE]

This shows that $A(y)$ is continuous at $y=0$ as well. ∎

5. Well-Separated Chebyshev-type cubature formulas

This section is devote to the proofs of Theorem 1.1 and Corollary 1.2.

Let $\Pi_{n,0,w}^{d}$ denote the set of all spherical polynomials $P$ of degree at most $n$ on $\mathbb{S}^{d-1}$ with $\int_{\mathbb{S}^{d-1}}P(x)w(x)\,d{\sigma}(x)=0$. Then $\Pi_{n,0,w}^{d}$ is a finite dimensional real Hilbert space equipped with the inner product of the space $L^{2}(\mathbb{S}^{d-1};w(x)d{\sigma}(x))$. Let $G_{n,w}(\cdot,\cdot)$ denote the reproducing kernel of the Hilbert space $\Pi_{n,0,w}^{d}$. Clearly, (1.5) is equivalent to the following

[TABLE]

Finally, we recall that $\|f\|_{1,w}=\int_{\mathbb{S}^{d-1}}|f(x)|w(x)\,d{\sigma}(x)$.

The proof of Theorem 1.1 follows closely the methods used in [6]. Indeed, applying Lemma 4.5 to the vector valued function $F(P;z):=\sum_{j=1}^{N}G_{n,w}(\mathbf{x}_{j}(P),z)$, $P\in\Pi_{n,0,w}^{d}$, $z\in\mathbb{S}^{d-1}$ and the open subset ${\Omega}:=\{P\in\Pi_{n,0,w}^{d}:\ \ \|\nabla_{0}P\|_{1,w}<1\}$ of the Hilbert space $\Pi_{n,0,w}^{d}$, and taking into account (5.1), we reduce to showing

Proposition 5.1**.**

For each integer $N\geq K_{w}M_{n,w}$, there exists a set $\{\mathbf{x}_{\alpha}\}_{{\alpha}\in\Lambda}$ of $N$ continuous functions $P\mapsto\mathbf{x}_{\alpha}(P)$ from the space $\Pi_{n,0,w}^{d}$ to $\mathbb{S}^{d-1}$ such that $\mathbf{x}_{\alpha}(P)\neq\mathbf{x}_{{\alpha}^{\prime}}(P)$ for $P\in\Pi_{n,0,w}^{d}$, ${\alpha},{\alpha}^{\prime}\in\Lambda$ and ${\alpha}\neq{\alpha}^{\prime}$, and such that

[TABLE]

where $\Lambda$ is an index set with cardinality $N$. If, in addition, $w\in L^{\infty}(\mathbb{S}^{d-1})$, then the set of points $\{\mathbf{x}_{\alpha}(P)\}_{{\alpha}\in\Lambda}$ is $c_{w}N^{-\frac{1}{d-1}}$-separated for every $P\in\Pi_{n,0,w}^{d}$; that is,

[TABLE]

for some positive constant $c_{w}$ depending only on $\|w\|_{\infty}$ and the doubling constant of $w$.

5.1. Proof of Proposition 5.1

Throughout the proof, $K_{w}$ denotes a sufficiently large constant depending only on the doubling constant of $w$. Set ${\delta}=2^{\frac{1}{s_{w}}}K_{w}^{-\frac{1}{s_{w}}}\in(0,1)$. By (1.3), for any $x\in\mathbb{S}^{d-1}$,

[TABLE]

Let $N,n\in{\mathbb{N}}$ be such that $N^{-\frac{1}{s_{w}}}=\frac{{\delta}}{n}$, where ${\delta}\in(0,1)$ is a small constant depending only on the doubling weight of $w$. According to Theorem 1.4, there exists a convex partition $\mathcal{R}=\{R_{1},\cdots,R_{M}\}$ of $\mathbb{S}^{d-1}$ with the properties that for each $1\leq j\leq M$, there exist a positive integer $k_{j}$ and a point $x_{j}\in R_{j}$ such that $w(R_{j})=\frac{k_{j}}{N}$ and $B(x_{j},\frac{c_{d}{\delta}}{n})\subset R_{j}\subset B(x_{j},\frac{c_{d}^{\prime}{\delta}}{n})$.

For each $1\leq j\leq M$, set r_{j}=\frac{{\delta}^{2}}{n}\bigl{(}Nw(R_{j}))\bigr{)}^{-\frac{1}{d-1}}. Since k_{j}r_{j}^{d-1}={\delta}^{d-1}\Bigl{(}\frac{{\delta}}{n}\Bigr{)}^{d-1}, there exists a set of $k_{j}$ points $x_{j,1},\cdots,x_{j,k_{j}}$ in the set $R_{j}$ which are $2r_{j}$-separated and satisfy $x_{j,1}=x_{j}$, and $B(x_{j,i},r_{j})\subset R_{j}$ for all $1\leq i\leq k_{j}$. If, in addition, $w\in L^{\infty}$, then

[TABLE]

which implies

[TABLE]

Let $P\in\Pi_{n,w,0}^{d}$. It is easily seen that $x\cdot\nabla_{0}P(x)=0$ for all $x\in\mathbb{S}^{d-1}$. Thus, according to Lemma 4.7, if $\nabla_{0}P(x_{j})\neq 0$, there exists a unique point $z_{j,P}\in R_{j}$ such that z_{j,P}\cdot\Bigl{(}\nabla_{0}P(x_{j})\Bigr{)}=\max_{z\in R_{j}}z\cdot\Bigl{(}\nabla_{0}P(x_{j})\Bigr{)}. Now for each $P\in\Pi_{n,0,w}^{d}$, we define

[TABLE]

We claim that for each $1\leq j\leq M$ and $1\leq i\leq k_{j}$, $P\mapsto\mathbf{x}_{j,i}(P)$ is a continuous function from $\Pi_{n,w,0}^{d}$ to $\mathbb{S}^{d-1}$. Indeed, by Lemma 4.7, we may write $\mathbf{x}_{j,i}(P)=A(\nabla_{0}P(x_{j}))$, where $A$ is defined in (4.5) with $x_{j},x_{j,i},\nabla_{0}P(x_{j})$ in place of $x_{0},z_{0}$ and $y$ respectively. According to Lemma 4.7 (ii), $\mathbf{x}_{j,i}(P)$ is a continuous function of $\nabla_{0}P(x_{j})$. On the other hand, however, since $\Pi_{n,w,0}^{d}$ is a finite dimensional vector space, the mapping $P\mapsto\nabla_{0}(P)(x_{j})$ is continuous on $\Pi_{n,0,w}^{d}$. This proves the claim.

Now we set $\Lambda:=\{(j,i):\ \ 1\leq j\leq M,\ \ 1\leq i\leq k_{j}\}$, and turn to the proof of (5.2). Assume that $P\in\Pi_{n,0,w}^{d}$ and $\|\nabla_{0}P\|_{1,w}=1$. Let $z_{j,P}$, $x_{j,i}$ be defined as above. For convenience, we also set

[TABLE]

Let $\delta\in(0,1)$ be a parameter to be specified later. Define

[TABLE]

Let $z_{j,\max}\in R_{j}$ be such that $P(z_{j,\max})=\max_{z\in R_{j}}P(z)$. We then split the sum \frac{1}{N}\sum_{j=1}^{M}\sum_{i=1}^{k_{j}}P\bigl{(}\mathbf{x}_{j,i}(P)\bigr{)} on the left hand side of (5.2) into the following four parts:

[TABLE]

Firstly, we estimate the first sum $\Sigma_{1}$ from below. Use Lemma 4.4, we obtain

[TABLE]

where we used the fact that $B(x_{j},\frac{c_{d}{\delta}}{n})\subset R_{j}\subset B(x_{j},\frac{c_{d}^{\prime}{\delta}}{n})$ and the doubling property of $w$ in the third step, and Lemma 4.1(ii) in the last step.

Secondly, we prove the following upper estimate of the second sum $\Sigma_{2}$:

[TABLE]

For simplicity, we write ${\gamma}_{j,i}(t)={\gamma}_{[z_{j,i,P},z_{j,\max}]}(t)$. If $\nabla_{0}P(x_{j})\neq 0$, then $z_{j,i,P}=z_{j,P}$, and hence, by Lemma 4.7 (i), the function $\nabla_{0}P(x_{j})\cdot{\gamma}_{j,i}(t)$ is decreasing on $[0,1]$, namely,

[TABLE]

On the other hand, note that if $\nabla_{0}P(x_{j})=0$, then $z_{j,i,P}=x_{j,i}$ and then the inequality (5.10) holds trivially. Thus,

[TABLE]

It follows that

[TABLE]

where the second step uses Lemma 4.1 (i). This proves the estimate (5.9).

Thirdly, we show the following upper estimate of $\Sigma_{3}$:

[TABLE]

For simplicity, we set ${\alpha}_{j,i}(t)={\gamma}_{[x_{j,i},z_{j,i,P}]}(t)$. Recall that $y_{j,i}={\alpha}_{j,i}(1-{\delta})$. We have

[TABLE]

It follows by Lemma 4.1 that

[TABLE]

This proves (5.11).

Fourthly, we estimate above the last sum $\Sigma_{4}$. Fix temporarily $1\leq j\leq M$ and a small parameter $\varepsilon\in(0,1)$. We consider the following three cases. If $\|\nabla_{0}P(x_{j})\|=0$, then $x_{j,i}=\mathbf{x}_{j,i}(P)=z_{j,i,P}=y_{j,i}$, and hence, $P(\mathbf{x}_{j,i}(P))-P(y_{j,i})=0$. If $\|\nabla_{0}P(x_{j})\|\geq\varepsilon>0$, then

[TABLE]

and hence in this case we also have $P(\mathbf{x}_{j,i}(P))-P(y_{j,i})=0$. Finally, if $0<\|\nabla_{0}P(x_{j})\|<\varepsilon$, then

[TABLE]

Putting the above together, we obtain

[TABLE]

Finally, combining the estimates (5.8), (5.9), (5.11), (5.12) with (5.7), we obtain

[TABLE]

To conclude the proof of (5.2), we just need to choose the parameters $\varepsilon,{\delta}$ small enough so that $0<\varepsilon,{\delta}<\frac{c_{w}}{4C_{w}}$.

It remains to show the separation property (5.3) under the additional condition $w\in L^{\infty}(\mathbb{S}^{d-1})$. To this end, we need the following simple lemma.

Lemma 5.2**.**

Let $z,\xi_{1},\xi_{2}\in\mathbb{S}^{d-1}$ be such that $\xi_{1}\cdot z=\xi_{2}\cdot z=0$. Given ${\theta}_{1},{\theta}_{2}\in(0,\frac{\pi}{4}]$, define

[TABLE]

where $\eta_{i}=z\cos{\theta}_{i}+\xi_{i}\sin{\theta}_{i}$. Then for any $t\in(0,1)$,

[TABLE]

For the moment, we take Lemma 5.2 for granted and proceed with the proof of (5.3). Without loss of generality, we may assume that $\|w\|_{\infty}=1$. Set $\Lambda=\{(j,i):\ \ 1\leq j\leq M,\ \ 1\leq i\leq k_{j}\}$. It is enough to prove that for $(j,i)\neq(j^{\prime},i^{\prime})\in\Lambda$ and every $P\in\Pi_{n,0,w}^{d}$ with $\|\nabla_{0}P\|_{1,2}\leq 1$,

[TABLE]

We first prove (5.13) for the case of $j=j^{\prime}$ and $1\leq i\neq i^{\prime}\leq k_{j}$. In this case, if $\nabla_{0}P(x_{j})=0$, then $\mathbf{x}_{j,i}(P)=x_{j,i}$, $\mathbf{x}_{j,i^{\prime}}(P)=x_{j,i^{\prime}}$, and hence by (5.5), $\mathtt{d}(\mathbf{x}_{j,i}(P),\mathbf{x}_{j,i^{\prime}}(P))\geq r_{j}\geq CN^{-\frac{1}{d-1}}$. If $\nabla_{0}P(x_{j})\neq 0$, then

[TABLE]

with

[TABLE]

which using Lemma 5.2 implies

[TABLE]

Next, we prove (5.13) for the case of $1\leq j\neq j^{\prime}\leq M$. Assume that $1\leq i\leq k_{j}$ and $1\leq i^{\prime}\leq k_{j^{\prime}}$. To prove (5.13), it suffices to show that $\mathtt{d}(\mathbf{x}_{j,i}(P),\partial R_{j})\geq c_{w}r_{j}\geq cN^{-\frac{1}{d-1}}$. Without loss of generality, we may assume that $\|\nabla_{0}P(x_{j})\|>0$ since otherwise $\mathbf{x}_{j,i}(P)=x_{j,i}$ and the claim is obvious. We first recall that $B(x_{j,i},r_{j})\subset R_{j}$, and

[TABLE]

with t_{j}=(1-{\delta})\min\Bigl{\{}\frac{\|\nabla_{0}P(x_{j})\|}{\varepsilon},1\Bigr{\}}. Hence, setting $\theta_{j,i}=\mathtt{d}(x_{j,i},z_{j,P})$, we have

[TABLE]

Let $h:R_{j}\to T_{x_{j,i}}$, $z\mapsto z-(z\cdot x_{j,i})x_{j,i}$ denote the orthogonal projection onto the tangential space $T_{x_{j,i}}:=\{y\in{\mathbb{R}}^{d}:\ \ y\cdot x_{j,i}=0\}$. According to the proof of Lemma 4.7, the set $S=h(R_{j})$ is strictly convex in the space $T_{x_{j,i}}$, $h(x_{j,i})=0$, and $\|h(z)\|=\sin(\mathtt{d}(x_{j,i},z))$ for $z\in R_{j}$. Furthermore,

[TABLE]

Write

[TABLE]

where

[TABLE]

Then,

[TABLE]

This implies that

[TABLE]

where

[TABLE]

Here we used the fact that the function $\frac{\sin t}{t}$ is decreasing on $[0,\frac{\pi}{2}]$. Since the function $d(u,\partial S)$ is concave on $S$, it follows that

[TABLE]

5.2. Proof of Lemma 5.2

For $t\in(0,1)$ and ${\theta}_{1},{\theta}_{2}\in(0,\frac{\pi}{4})$,

[TABLE]

Thus,

[TABLE]

5.3. Proof of Corollary 1.2

The upper estimate,

[TABLE]

follows directly from Theorem 1.1. It remains to show the matching lower estimate.

Assume that $\Lambda=\{\omega_{j}\}_{j=1}^{N}$ is a set of $N$ distinct nodes on $\mathbb{S}^{d-1}$ such that

[TABLE]

According to [9, Lemma 4.6], given a positive integer $\ell\geq s_{w}+d+1$, there exists a nonnegative algebraic polynomial $P_{n}$ of degree at most $n/4$ on $[-1,1]$ such that

[TABLE]

where ${\langle}x,y{\rangle}$ denotes the dot product of $x,y\in{\mathbb{R}}^{d}$. Using (5.14) and [9, Theorem 4.2] we have that for $p=1,2$,

[TABLE]

where $\|\cdot\|_{p,w}$ denotes the Lebesgue $L^{p}$-norm defined with respect to the measure $w(x)\,d{\sigma}_{d}(x)$ on $\mathbb{S}^{d-1}$. Since the norm $\|\cdot\|_{\ell^{p}}$ is a decreasing function in $p>0$, it follows that

[TABLE]

Next, by [9, Corollary 3.4], $\|P_{n}({\langle}x,\cdot{\rangle})\|_{p,w}\sim\|P_{n}({\langle}x,\cdot{\rangle})\|_{p,w_{n}}$ for $1\leq p<\infty$, where $w_{n}(x)=n^{d-1}\int_{B(x,n^{-1})}w(y)\,d{\sigma}_{d}(y)$ for $x\in\mathbb{S}^{d-1}$ and $n=1,2,\cdots.$ This combined with (5.15) implies that for $1\leq p<\infty$,

[TABLE]

Thus, using (5.17) and (5.16) , we deduce that for any $x\in\mathbb{S}^{d-1}$,

[TABLE]

It follows that

[TABLE]

Acknowledgement

We would like to express sincere gratitude to Professor Ron Peled from the Tel Aviv University for kindly pointing out to us several very helpful references on Chebyshev-type cubature formulas.