Limit theorems for random polytopes with vertices on convex surfaces

TL;DR

This paper establishes limit theorems, variance bounds, and normal approximation results for the intrinsic volumes of random polytopes formed by points on convex surfaces, advancing understanding of their probabilistic geometric properties.

Contribution

It provides new variance bounds, strong laws, and central limit theorems for intrinsic volumes of convex hulls of boundary points, using Stein's method and surface body estimates.

Findings

Variance bounds for intrinsic volumes

Strong laws of large numbers

Central limit theorems with normal approximation

Abstract

The random polytope $K_n$, defined as the convex hull of $n$ points chosen uniformly at random on the boundary of a smooth convex body, is considered. Proofs for lower and upper variance bounds, strong laws of large numbers and central limit theorems for the intrinsic volumes of $K_n$ are presented. A normal approximation bound from Stein's method and estimates for surface bodies are among the involved tools.

Equations

$$\operatorname{\mathbf{Var}}[V_{\ell}(K_{n})]=\Theta\bigl{(}n^{-\frac{d+3}{d-1}}\bigr{)},\quad\ell\in\{1,\dots,d\}.$$

$$\mathbf{P}\Bigl{(}\lim_{n\to\infty}\bigl{(}V_{\ell}(K)-V_{\ell}(K_{n})\bigr{)}\cdot n^{\frac{2}{d-1}}=c_{d,\ell,K}\Bigr{)}=1,\quad\ell\in\{1,\dots,d\},$$

$$W_{\ell }(K_n):=\frac{V_{\ell }(K_n)-E[V_{\ell }(K_n)]}{\sqrt{\mathrm{Var}[V_{\ell }(K_n)]}},\ell \in \{1,\dots ,d\}.$$

$$\underset{u\in \mathbb{R}}{\sup }|\mathbf{P}(W_{\ell }(K_n)\le u)-\mathbf{P}(N\le u)|\ll n^{-\frac{1}{2}}(\log n{)}^{3+\frac{6}{d-1}},\ell \in \{1,\dots ,d\},$$

$$h_K(u)=\sup \{⟨x,u⟩:x\in K\},u\in {\text{SS}}^{d-1}.$$

$$V_{\ell }(K):=\left(\genfrac{}{}{0ex}{}{d}{\ell }\right)\frac{{\kappa }_d}{{\kappa }_{\ell }{\kappa }_{d-\ell }}{\int }_{G(d,\ell )}{\mathrm{vol}}_{\ell }(K|L){\nu }_{\ell }(\mathrm{d}L).$$

$$v(x):=\min \{V_d(K\cap H):H\text{is a half space in}{\mathbb{R}}^d\text{containing}x\}.$$

$$K(v\ge t):=\{x\in K:v(x)\ge t\}$$

$$K(t)=K(v\le t):=\{x\in K:v(x)\le t\}.$$

$$s(x):=\min \{{\mathcal{H}}^{d-1}(\partial K\cap H):H\text{is a half space in}{\mathbb{R}}^d\text{containing}x\}.$$

$$K(s\ge t):=\{x\in K:s(x)\ge t\}.$$

$$K(s\le t):=\{x\in K:s(x)\le t\}.$$

$${\mathrm{Vis}}_z(t):=\{x\in K(s\le t):[x,z]\cap K(s\ge t)=\varnothing \},$$

$$\mathbf{P}(K(s\ge {\tau }_n)⊈K_n)\le n^{-\alpha },$$

$${\tau }_n:=c_{\alpha }\frac{\log n}{n}.$$

$$b_{Q_x}(y)=\frac{1}{2}\sum _{i,j}{f}_{ij}^{(x)}(0)y_i{y}_j$$

$$(1+\delta {)}^{-1}{b}_{Q_x}(y)\le f^{(x)}(y)\le (1+\delta )b_{Q_x}(y)$$

$$c_1^{-1}{\epsilon }^{(d-1)/(d+1)}\le {\mathcal{H}}^{d-1}(C^K\cap \partial K)\le c_1{\epsilon }^{(d-1)/(d+1)}$$

$$c_2^{-1}{\epsilon }^{(d+1)/(d-1)}\le V_d({\tilde{C}}^K)\le c_2{\epsilon }^{(d+1)/(d-1)}.$$

$$\nu_{\ell}(\{L\in G(d,\ell):\sphericalangle(z,L)\leq a\})=\Theta\bigl{(}a^{d-\ell}\bigr{)},\quad\ell\in\{1,\dots,d\}.$$

$$d_K({\mathbf{P}}_X,{\mathbf{P}}_Y)=\underset{u\in \mathbb{R}}{\sup }|\mathbf{P}(X\le u)-\mathbf{P}(Y\le u)|.$$

$$D_{i}f(x):=f(x)-f(x^i)\text{and}{D}_{i,j}f(x):=f(x)-f(x^i)-f(x^j)+f(x^{ij}),$$

$$X_i^A:=\{\begin{array}{ll}{X}_i& :i\notin A,\\ X_i^{\prime }& :i\in A.\end{array}$$

$${\gamma }_1$$

$${\gamma }_2$$

$${\gamma }_3$$

$${\gamma }_4$$

$${\gamma }_5$$

$$d_{K}\biggl{(}\frac{W}{\sqrt{\operatorname{\mathbf{Var}}[W]}},N\biggr{)}\ll\frac{\sqrt{n}}{\operatorname{\mathbf{Var}}[W]}(\sqrt{n^{2}\gamma_{1}}+\sqrt{n\gamma_{2}}+\sqrt{\gamma_{3}})+\frac{n}{(\operatorname{\mathbf{Var}}[W])^{\frac{3}{2}}}\gamma_{4}+\frac{n}{(\operatorname{\mathbf{Var}}[W])^{2}}\gamma_{5}.$$

$$E=\{z\in {\mathbb{R}}^d:z_d\ge z_1^2+\cdots +z_{d-1}^2\}.$$

$$\{\lambda z\in {\mathbb{R}}^d:\lambda \ge 0,z\in S\}\supseteq 2E\cap H(e_d,1).$$

$$\tilde{b}:{\mathbb{R}}^{d-1}\to \partial E,y\mapsto (y,b_E(y)).$$

Full text

Keywords. Central limit theorem, intrinsic volume, random polytope, stochastic geometry, surface body, variance.

MSC (2010). 52A22, 60D05, 60F05.

1 Introduction and main results

For fixed $d\geq 2$, let $\mathcal{K}^{2}_{+}$ be the set of convex bodies in $\mathbb{R}^{d}$ which have a twice differentiable boundary with everywhere positive Gaussian curvature. Let $K\in\mathcal{K}^{2}_{+}$ be a convex body. We denote by $\mathcal{H}^{d-1}$ the $(d-1)$-dimensional Hausdorff measure on $\partial K$, normalized such that $\mathcal{H}^{d-1}(\partial K)=1$. For $n\geq d+1$, we choose random points $X_{1},\ldots,X_{n}$ from $\partial K$, independently and according to $\mathcal{H}^{d-1}$. We denote by $K_{n}$ the convex hull of $X_{1},\ldots,X_{n}$. This means that $K_{n}$ is a random polytope having its vertices on the boundary of $K$. The interest of this paper is about the intrinsic volumes $V_{\ell}(K_{n})$ of $K_{n}$, $\ell\in\{1,\ldots,d\}$. The importance of these functionals is well-known and arises from convex and integral geometry. Indeed, as Hadwiger’s theorem states, they form (together with the Euler characteristic) a basis of the vector space of all motion invariant and continuous valuations on convex bodies. With this paper we provide lower and upper variance bounds, strong laws of large numbers and central limit theorems for $V_{\ell}(K_{n}),\,\ell\in\{1,\ldots,d\}$, filling some gaps that remain in the study of these objects.

Intrinsic volumes have been studied extensively in the alternative setting of random polytopes that arise as convex hulls of points chosen uniformly at random inside a fixed convex body. Results concerning the expectation of $V_{\ell}(K_{n})$, $\ell\in\{1,\ldots,d\}$, have been studied, for example, by Reitzner [10], variance bounds can be found in Böröczky, Fodor, Reitzner and Vígh [4] and Bárány, Fodor and Vígh [1], and central limit theorems were treated in Reitzner [11], Vu [17], Lachièze-Rey, Schulte and Yukich [7] and Thäle, Turchi and Wespi [16]. More details can be found in the references therein.

On the other hand, the approximation of a convex body $K$, by means of a sequence of random polytopes $K_{n}$, is improved whenever the vertices of $K_{n}$ are restricted to lie on the boundary of $K$, therefore making it a model worth studying. Indeed, in this framework the expectations of $V_{\ell}(K_{n})$, $\ell\in\{1,\ldots,d\}$, have been studied, for example, by Buchta, Müller and Tichy [5], Reitzner [8], Schütt and Werner [14] and Böröczky, Fodor and Hug [3]. However, more detailed informations are only known about the distribution of the volume $V_{d}(K_{n})$. In particular, an upper variance bound was found by Reitzner [9] and a lower variance bound together with concentration inequalities by Richardson, Vu and Wu [12]. Only recently, Thäle [15] obtained a quantitative central limit theorem for $V_{d}(K_{n})$ based on Stein’s method.

Our first aim is to generalize the results obtained in [9, 12] to $V_{\ell}(K_{n})$, $\ell\in\{1,\ldots,d\}$. In fact, we prove lower variance bounds following the ideas of [1, 11, 12] and upper variance bounds in the manner of [1], making use of the Efron-Stein jackknife inequality from [9]. In particular, the upper variance bounds imply strong laws of large numbers as in [1]. Secondly, we prove quantitative central limit theorems for $V_{\ell}(K_{n})$, $\ell\in\{1,\ldots,d\}$, using a normal approximation bound obtained in [6], extending the result of [15].

We now introduce some notation in order to present our results. Let $(a_{n})_{n\in\mathbb{N}}$ and $(b_{n})_{n\in\mathbb{N}}$ be two sequences of real numbers. We write $a_{n}\ll b_{n}$ (or $a_{n}\gg b_{n}$) if there exist a constant $c\in(0,\infty)$ and a positive number $n_{0}$ such that $a_{n}\leq c\,b_{n}$ (or $a_{n}\geq c\,b_{n}$) for all $n\geq n_{0}$. Furthermore, $a_{n}=\Theta(b_{n})$ means that $b_{n}\ll a_{n}\ll b_{n}$.

Our first result concerns asymptotic lower and upper bounds, respectively, for the variances of the intrinsic volumes.

Theorem 1.1**.**

Let $K\in\mathcal{K}_{+}^{2}$ and choose $n$ random points on $\partial K$ independently and according to the probability distribution $\mathcal{H}^{d-1}$. Then,

[TABLE]

Based on a result stated in [8, Theorem 1] concerning the behaviour of $V_{\ell}(K)-\operatorname{\mathbf{E}}[V_{\ell}(K_{n})]$, the upper variance bounds of Theorem 1.1 imply strong laws of large numbers.

Theorem 1.2**.**

In the set-up of Theorem 1.1, it holds that

[TABLE]

for some positive constants $c_{d,\ell,K}$ that depend on $d,\ell$ and $K$.

The constants $c_{d,\ell,K}$ appear in an explicit form in [8, Theorem 1] and can be expressed in form of integrals of the principal curvatures of $K$.

Next, we introduce the standardized intrinsic volume functionals, defined by

[TABLE]

We prove the following central limit theorems for such functionals.

Theorem 1.3**.**

In the set-up of Theorem 1.1, it holds that

[TABLE]

where $N$ is a standard Gaussian random variable. In particular, $W_{\ell}(K_{n})$ converges in distribution to $N$, as $n\to\infty$.

Note that the rate of convergence in Theorem 1.3 does not depend on $\ell$. Moreover, the same rate of convergence was already obtained in [15] for the case $\ell=d$.

The paper is organised as follows. In Section 2 we introduce the notation and recall some background material from convex geometry, results concerning the surface and floating bodies and the normal bound from [6] that will be used in the proof of Theorem 1.3. In Section 3 we present the geometric construction needed for the proof of the lower bounds of Theorem 1.1 and the proof itself. In Section 4 we prove the upper bounds of Theorem 1.1 by means of the Efron-Stein jackknife inequality and we also prove Theorem 1.2, which directly follows from the former. Finally, in Section 5 we give the proof of Theorem 1.3.

2 Background material

2.1 General notation

The closed Euclidean ball of radius $r$ centred at $x\in\mathbb{R}^{d}$ is denoted by $B^{d}(x,r)$, and $B^{d}=B^{d}(0,1)$ stands for the centred Euclidean unit ball. The boundary of $B^{d}$ is indicated with $\SS^{d-1}$. Moreover, the volume of $B^{d}$ is denoted by $\kappa_{d}=\pi^{d/2}\Gamma(1+{\frac{d}{2}})^{-1}$. For a finite set $A=\{x_{1},\ldots,x_{n}\}\subset\mathbb{R}^{d}$, the convex hull of $A$ is denoted by $[x_{1},\ldots,x_{n}]$. The vectors $e_{1},\dots,e_{d}$ represent the standard orthonormal basis of $\mathbb{R}^{d}$. We indicate with $\sphericalangle(u,v)$ the angle between two vectors $u,v$ $\in\mathbb{R}^{d}$. For a linear subspace $V$ of $\mathbb{R}^{d}$, we define $\sphericalangle(u,V)\coloneqq\inf\{\sphericalangle(u,v):v\in V\}$. Given a subset $U\in\mathbb{R}^{d},$ its projection onto $\mathbb{R}^{d-1}$ is denoted by $\mathrm{proj}_{\mathbb{R}^{d-1}}U=\{x\in\mathbb{R}^{d-1}:(x,y)\in U\text{ for some }y\in\mathbb{R}\}$. For a function $f\colon\mathbb{R}^{d}\to\mathbb{R}$, we say $f\in\mathcal{C}^{2}$ if it is twice differentiable with continuous second order partial derivatives.

Let $u\in\mathbb{R}^{d}$ and $h\in\mathbb{R}$. We denote by $H(u,h)$ the hyperplane $\{x\in\mathbb{R}^{d}:\langle x,u\rangle=h\}$. The corresponding halfspace $\{x\in\mathbb{R}^{d}:\langle x,u\rangle\geq h\}$ is denoted by $H^{+}(u,h)$. Often one describes a convex body by its support function. The support function of $K$ is defined by

[TABLE]

Since $K\in\mathcal{K}_{+}^{2}$, there exists a unique unit outward normal $u_{x}$ for each $x\in\partial K$. The intersection of $K$ with $H^{+}(u_{x},h_{K}(u_{x})-h)$ is denoted by $C^{K}(x,h)$. We call $C^{K}(x,h)$ a cap of $K$ at $x$ of height $h$. A cap $C^{K}$ is called an $\varepsilon$-cap if $V_{d}(C^{K})=\varepsilon$, where $V_{d}(\cdot)$ denotes the $d$-dimensional volume. Analogously, a cap $C^{K}$ with $\mathcal{H}^{d-1}(C^{K}\cap\partial K)=\varepsilon$ is called an $\varepsilon$-boundary cap.

Let $\ell\in\{0,\dots,d\}$. We denote by $G(d,\ell)$ the Grassmannian of all $\ell$-dimensional linear subspaces of $\mathbb{R}^{d}$, which is supplied with the unique Haar probability measure $\nu_{\ell}$, see [13]. For $L\in G(d,\ell)$, we write $\operatorname{vol}_{\ell}(K|L)$ to indicate the $\ell$-dimensional Lebesgue measure of the orthogonal projection of $K$ onto $L$. Then, the $\ell\text{-th}$ intrinsic volume of a convex body $K$ can be defined as

[TABLE]

In particular, $V_{d}(K)$ is the ordinary volume (Lebesgue measure), $V_{d-1}(K)$ is half of the surface area, $V_{1}(K)$ is a constant multiple of the mean width and $V_{0}(K)$ is the Euler-characteristic of $K$.

We define the function $v:K\to\mathbb{R}$ by

[TABLE]

Then, the set

[TABLE]

is called the floating body of $K$ with parameter $t>0$. The wet part of $K$ is defined by

[TABLE]

In a similar way, we define the function $s:K\to\mathbb{R}$ by

[TABLE]

The surface body of $K$ with parameter $t>0$ is defined by

[TABLE]

Analogously, we set

[TABLE]

We define the visibility region (with respect to $s$) of a point $z\in\partial K$ with parameter $t>0$ as

[TABLE]

where $[x,z]$ denotes the closed line segment which connects $x$ and $z$.

We use the convention that constants with the same subscript may differ from section to section.

2.2 Geometric tools

The concept of the surface body is convenient in view of Lemma 2.1, which clarifies its connection with the random polytope $K_{n}$.

Lemma 2.1**.**

[12, Lemma 4.2]* For all $\alpha\in(0,\infty)$, there exists a constant $c_{\alpha}\in(0,\infty)$ only depending on $\alpha$ such that*

[TABLE]

where

[TABLE]

In the following, we present some well-known geometric results in order to keep our presentation reasonably self-contained. For every point $x\in\partial K$, there exists a paraboloid $Q_{x}$, given by a quadratic form $b_{Q_{x}}$, osculating at $x$. The following precise description of the local behaviour of the boundary of a convex body $K\in\mathcal{K}_{+}^{2}$ is due to Reitzner [8].

Lemma 2.2**.**

[8, Lemma 6]* Let $K\in\mathcal{K}_{+}^{2}$ and choose $\delta>0$ sufficiently small. Then, there exists a $\lambda>0$, only depending on $\delta$ and $K$, such that for each $x\in\partial K$ the following holds. Identify the hyperplane tangent to $K$ at $x$ with $\mathbb{R}^{d-1}$ and $x$ with the origin. The $\lambda$-neighbourhood $U^{\lambda}$ of $x$ in $\partial K$ defined by $\operatorname{proj}_{\mathbb{R}^{d-1}}U^{\lambda}=\lambda B^{d-1}$ can be represented by a convex function $f^{(x)}(y)\in\mathcal{C}^{2}$, i.e., $(y,f^{(x)}(y))\in\partial K$ for $y\in\lambda B^{d-1}$. Denote by $f_{ij}^{(x)}(0)$ the second partial derivatives of $f^{(x)}$ at the origin. Then,*

[TABLE]

and it holds that

[TABLE]

for $y\in\lambda B^{d-1}$.

In the next Lemma we state two well-known relations regarding $\varepsilon$-caps and $\varepsilon$-boundary caps.

Lemma 2.3**.**

[12, Lemma 6.2]* For a given $K\in\mathcal{K}_{+}^{2}$, there exist constants $\varepsilon_{0},c_{1},c_{2}>0$ such that for all $0<\varepsilon<\varepsilon_{0}$ we have that for any $\varepsilon$-cap $C^{K}$ of $K$,*

[TABLE]

and for any $\varepsilon$-boundary cap $\widetilde{C}^{K}$ of K,

[TABLE]

For the next geometrical Lemma we assume that $\varepsilon$ is sufficiently small.

Lemma 2.4**.**

[18, Lemma 6.2]* Let $x$ be a point on the boundary of $K$ and $D(x,\varepsilon)$ the set of all points on the boundary which are of distance at most $\varepsilon$ to $x$. Then, the convex hull of $D(x,\varepsilon)$ has volume at most $c_{3}\varepsilon^{d+1}$, where $c_{3}>0$ is a constant.*

The following result is known as the economic cap covering theorem, see [1, 2].

Proposition 2.5**.**

[1, Theorem 4]* Assume that $K$ is a convex body with unit volume and let $0<t<t_{0}=(2d)^{-2d}$. Then, there are caps $C_{1},\dots,C_{m}$ and pairwise disjoint convex sets $C^{\prime}_{1},\dots,C^{\prime}_{m}$ such that $C^{\prime}_{i}\subset C_{i}$ for each $i$, and*

$\bigcup_{i=1}^{m}C^{\prime}_{i}\subset K(t)\subset\bigcup_{i=1}^{m}C_{i}$, 2. 2.

$V_{d}(C^{\prime}_{i})\gg t$* and $V_{d}(C_{i})\ll t$ for each $i$,* 3. 3.

for each cap $C$ with $C\cap K(v>t)=\emptyset$, there is a $C_{i}$ containing $C$.

We conclude this section with a statement about the measure of the set of linear subspaces of $\mathbb{R}^{d}$ that form a small angle with a fixed vector, which will be useful later.

Lemma 2.6**.**

[1, Lemma 1]* For fixed $z\in\SS^{d-1}$ and small $a>0$,*

[TABLE]

2.3 Bound for normal approximation

Let $(\Omega,\mathcal{A},\mathbf{P})$ be a fixed probability space. We indicate with $\operatorname{\mathbf{E}}$ the expectation and with $\operatorname{\mathbf{Var}}$ the variance with respect to $\mathbf{P}$. Let $X$ and $Y$ be two real-valued random variables defined on $(\Omega,\mathcal{A},\mathbf{P})$ with distributions $\mathbf{P}_{X}$ and $\mathbf{P}_{Y}$, respectively. The Kolmogorov distance between $\mathbf{P}_{X}$ and $\mathbf{P}_{Y}$ is defined by

[TABLE]

With a slight abuse of notation, we write $d_{K}(X,Y)$ to indicate $d_{K}(\mathbf{P}_{X},\mathbf{P}_{Y})$. It is important to recall that the Kolmogorov distance is a metrization of the convergence in distribution, i.e., given a sequence of random variables $(X_{n})_{n\in\mathbb{N}}$ and another random variable $Y$ such that $\lim\limits_{n\to\infty}d_{K}(X_{n},Y)=0$, then $(X_{n})_{n\in\mathbb{N}}$ converges in distribution to $Y$.

Let $S$ be a Polish space. Consider a function $f\colon\cup_{k=1}^{n}S^{k}\to\mathbb{R}$ that acts on the point configurations of at most $n\in\mathbb{N}$ points of $S$. Let $f$ be measurable and symmetric, i.e., invariant under permutations of the arguments. In the setting of this paper, $S$ is the boundary of a smooth convex body, while $f$ is an intrinsic volume of the convex hull of its arguments. Given a point $x=(x_{1},\ldots,x_{h})\in\cup_{k=1}^{n}S^{k}$, we indicate with $x^{i}$ the vector obtained from $x$ by removing its $i\text{-th}$ coordinate, namely $x^{i}\coloneqq(x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{h})$. Analogously, we define $x^{ij}\coloneqq(x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{j-1},x_{j+1},\ldots,x_{h})$.

We now define the first- and second-order difference operators, applied to $f$, as

[TABLE]

respectively. We indicate with $X=(X_{1},\ldots,X_{n})$ a random vector of elements of $S$ on $(\Omega,\mathcal{A},\mathbf{P})$. Let $X^{\prime}$ and $\tilde{X}$ be independent copies of $X$. A vector $Z=(Z_{1},\ldots,Z_{n})$ is called a recombination of $\{X,X^{\prime},\tilde{X}\}$, whenever $Z_{i}\in\{X_{i},X_{i}^{\prime},\tilde{X}_{i}\}$ for every $i\in\{1,\ldots,n\}$. For a subset $A\subseteq\{1,\ldots,n\}$ of the index set, we write $X^{A}=(X_{1}^{A},\ldots,X_{n}^{A})$ with

[TABLE]

In order to rephrase the normal approximation bound from [6], it is convenient to define the following quantities, namely,

[TABLE]

where the suprema in the definitions of $\gamma_{1}$ and $\gamma_{2}$ run over all combinations of vectors $(Y,Y^{\prime},Z,Z^{\prime})$ or $(Y,Z,Z^{\prime})$ that are recombinations of $\{X,X^{\prime},\tilde{X}\}$.

Proposition 2.7**.**

[6, Theorem 5.1]* Let $W:=f(X_{1},\dots,X_{n})$ and assume that $\operatorname{\mathbf{E}}[W]=0$ and $\operatorname{\mathbf{E}}[W^{2}]<\infty$. Moreover, let $N$ be a standard Gaussian random variable. Then, the following bound for the normal approximation holds:*

[TABLE]

3 Lower variance bounds

3.1 Auxiliary geometric construction

The following geometrical construction is taken from [12, Section 3.1]. Let $E$ be the standard paraboloid given by

[TABLE]

We construct a simplex $S$ in $C^{E}(0,1)$ in the following way. The base is a regular simplex whose vertices $v_{1},\ldots,v_{d}$ lie on $\partial E\cap H(e_{d},1/(3(d-1)^{2}))$ while $v_{0}=(0,\ldots,0)$ is the apex of $S$. Notice that the ball $2E\cap H(e_{d},1)$ has radius $\sqrt{2}$, while the inradius of the base of the simplex is $1/(\sqrt{3}(d-1)^{2})$ and therefore, $\{\lambda z\in\mathbb{R}^{d}:\lambda\geq 0,z\in S\}\cap H(e_{d},1)$ has inradius $3(d-1)^{2}/(\sqrt{3}(d-1)^{2})=\sqrt{3}$. In particular, this implies that

[TABLE]

For $i\in\{0,1,\ldots,d\}$, let $v_{i}^{\prime}$ be the orthogonal projection of $v_{i}$ onto $\mathrm{span}\{e_{1},\ldots,e_{d-1}\}$. Consider $B_{0}\coloneqq B^{d-1}(v_{0}^{\prime},r)\subseteq\mathbb{R}^{d-1}$ and $B_{i}\coloneqq B^{d-1}(v_{i}^{\prime},r^{\prime})\subseteq\mathbb{R}^{d-1}$, $i\in\{1,\dots,d\}$, for some radii $r$ and $r^{\prime}$ to be chosen later. Let $b_{E}$ be the quadratic form associated with $E$, i.e., $b_{E}(y)=\lVert y\rVert^{2}$ for $y\in\mathbb{R}^{d-1}$. For $i\in\{0,\ldots,d\}$, we can define the lift $B_{i}^{\prime}\coloneqq\tilde{b}(B_{i})$ on $\partial E$ of the sets $B_{i}$, where $\tilde{b}$ indicates the mapping

[TABLE]

Note that, if $r$ and $r^{\prime}$ are small enough, then, by continuity, for any $(d+1)\text{-tuple}$ of points $x_{i}\in B_{i}^{\prime}$, the following still holds,

[TABLE]

Then, we extend the aforementioned argument to arbitrary caps of $\partial K$. For each point $x\in\partial K$, we consider the approximating paraboloid $Q_{x}$ of $K$ at $x$. Let $T_{x}(K)$ be the tangent space of $K$ at the point $x$. The space $T_{x}(K)$ can be identified with $\mathbb{R}^{d-1}$ having $x$ as its origin. Then, there exists a unique affine map $A_{x}$ such that $A_{x}(C^{E}(0,1))=C^{Q_{x}}(x,h)$ while mapping the coordinate axes onto the coordinate axes of $T_{x}(K)\times\mathbb{R}$. We define $D_{i}(x)\coloneqq A_{x}(B_{i})$, $i\in\{0,\dots,d\}$. Then, it is true that $\operatorname{vol}_{d-1}(D_{i}(x))=c_{1}\,h^{\frac{d-1}{2}}$ for a constant $c_{1}>0$. We define now $D_{i}^{\prime}(x)\coloneqq\tilde{f}^{(x)}(D_{i}(x))$, for a neighbourhood $U\subseteq T_{x}(K)$ of $x$,

[TABLE]

Since $K\in\mathcal{K}_{+}^{2}$, there exist positive lower and upper bounds for the curvature. Thus, due to the curvature bounds of $K$, it holds that

[TABLE]

where $c_{K}$ and $C_{K}$ are positive constants depending only on $K$.

By continuity, if every $x_{i}$ belongs to a ball $B^{d}(v_{i},\eta)$, (2) is preserved whenever $\eta>0$ is small enough. Moreover, we can choose $r$ and $r^{\prime}$ to be small enough such that for every $x\in\partial K$ and every $i\in\{0,\ldots,d\}$, $D^{\prime}_{i}(x)\subseteq A_{x}(B^{d}(v_{i},\eta))$. Indeed, define for $\varepsilon>0$ and every $i\in\{0,\ldots,d\}$, the set $U_{i}=\{(x,y)\in\mathbb{R}^{d}:x\in B^{d-1}(\operatorname{proj}_{\mathbb{R}^{d-1}}v_{i},\eta/2),y\in[(1+\varepsilon)^{-1}b_{E}(x),(1+\varepsilon)b_{E}(x)]\}$. If $\varepsilon$ is small enough, then $U_{i}\subseteq B^{d}(v_{i},\eta)$. Using Lemma 2.2, we can take $h$ small enough such that $(1+\varepsilon)^{-1}b_{Q_{x}}(y)\leq f^{(x)}(y)\leq(1+\varepsilon)b_{Q_{x}}(y)$. In particular, if we choose $r,r^{\prime}<\eta/2$, then $D_{i}^{\prime}(x)\subseteq A_{x}(U_{i})\subseteq A_{x}(B^{d}(v_{i},\eta))$. As a consequence, we have, for $x_{i}\in D_{i}^{\prime}(x)$,

[TABLE]

where the last inclusion holds whenever $h\leq h_{0}$ for $h_{0}$ sufficiently small. Therefore, from now on $r,r^{\prime}$ and $h_{0}$ are chosen such that the previous argument holds true.

3.2 Proof of the lower bounds

In this section we combine tools from [1, 11, 12]. Let $K\in\mathcal{K}_{+}^{2}$ and $X_{1},\dots,X_{n}$ be independent random points that are chosen from $\partial K$ according to the probability distribution $\mathcal{H}^{d-1}$. Due to [11, Lemma 13], we can choose $n$ points $y_{1},\dots,y_{n}\in\partial K$ and corresponding disjoint caps of $K$, namely, $C^{K}(y_{j},h_{n})$ for $j\in\{1,\dots,n\}$, with h_{n}=\Theta\bigl{(}n^{-\frac{2}{d-1}}\bigr{)}. For all $i\in\{0,\dots,d\}$ and $j\in\{1,\dots,n\}$, we define the sets $\{D_{i}(y_{j})\}$ and $\{D_{i}^{\prime}(y_{j})\}$ as in Section 3.1. Let $A_{j}$, $j\in\{1,\dots,n\}$, be the event that exactly one random point is contained in each $D^{\prime}_{i}(y_{j})$, $i\in\{0,\dots,d\}$, and every other point is outside of $C^{K}(y_{j},h_{n})\cap\partial K$.

Lemma 3.1**.**

[12, Section 3.2]* For $n$ large enough, and all $j\in\{1,\dots,n\}$, there exists a constant $c\in(0,1)$ such that $\mathbf{P}(A_{j})\geq c$.*

Proof.

The probability of the event $A_{j}$ is

[TABLE]

Combining Lemma 2.3, [11, Lemma 13] and Equation 3, we obtain

[TABLE]

where all constants are positive. ∎

Let $\mathcal{F}$ be the $\sigma$-field generated by the positions of all $X_{1},\dots,X_{n}$ except those which are contained in $D^{\prime}_{0}(y_{j})$ with $\boldsymbol{1}_{A_{j}}=1$. Assume that $\boldsymbol{1}_{A_{j}}=\boldsymbol{1}_{A_{k}}=1$ for some $j,k\in\{1,\dots,n\}$ and without loss of generality that $X_{j}$ and $X_{k}$ are the points in $D^{\prime}_{0}(y_{j})$ and $D^{\prime}_{0}(y_{k})$. By Equation 4, it is not possible that there is an edge between $X_{j}$ and $X_{k}$. Therefore, the change of the intrinsic volume affected by moving $X_{j}$ within $D^{\prime}_{0}(y_{j})$ is independent of the change of the intrinsic volume of moving $X_{k}$ within $D^{\prime}_{0}(y_{k})$. As a consequence, we obtain

[TABLE]

where the variances $\operatorname{\mathbf{Var}}_{j}[\cdot]$ are taken over $X_{j}\in D^{\prime}_{0}(y_{j})$.

For $j\in\{1,\dots,n\}$ and $i\in\{0,\dots,d\}$, let $z_{j}^{i}$ be an arbitrary point in $D^{\prime}_{i}(y_{j})$. We indicate with $N_{j}$ the normal cone of the simplex $[z_{j}^{0},\dots,z_{j}^{d}]$ at vertex $z_{j}^{0}$. Let $S_{j}$ be the cone with base $H(u_{z^{0}_{j}},h_{K}(u_{z_{j}^{0}})-h_{n})\cap 2Q_{x}$ and vertex $z_{j}^{0}$. Note that $u_{z_{j}^{0}}$ is the unique unit outer normal of $K$ at $z_{j}^{0}$. The corresponding normal cone of $S_{j}$ at $z_{j}^{0}$ is denoted by $\bar{N}_{j}$. Moreover, the angular aperture of $S_{j}$ at $z_{j}^{0}$ is at most $c^{\prime}_{K}\sqrt{h_{n}}$, where $c^{\prime}_{K}>0$ is a constant that depends on $K$. Because of this and Equation 4, we can find sets $\Sigma_{j}$ such that

[TABLE]

We fix $j\in\{1,\dots,n\}$ and $z_{j}^{i}\in D^{\prime}_{i}(y_{j})$ for all $i\in\{1,\dots,d\}$. Let $F_{j}:=[z_{j}^{1},\dots,z_{j}^{d}]$ and define

[TABLE]

Lemma 3.2**.**

Let $j\in\{1,\dots,n\}$ and let $X_{j}$ be a point chosen with respect to the normalized Hausdorff measure restricted to $D^{\prime}_{0}(y_{j})$. Then,

[TABLE]

Proof.

Note that $[X_{j},F_{j}]|L$ is a simplex in $L\in G(d,\ell)$ with base $F_{j}|L$ and additional point $X_{j}|L$. As a consequence, the height of $[X_{j},F_{j}]|L$ is proportional to $h_{n}$ and

[TABLE]

where $L\in G(d,\ell)$ with $L\cap\Sigma_{j}\neq\emptyset$. Thus,

[TABLE]

Due to Lemma 2.6 and Equation 5, it follows

[TABLE]

Therefore, we obtain

[TABLE]

Let $X_{j}^{1}$ and $X_{j}^{2}$ be independent copies of $X_{j}$, then

[TABLE]

since the heights of $X_{j}^{1}|L$ and $X_{j}^{2}|L$ are different with probability 1. Using h_{n}=\Theta\bigl{(}n^{-\frac{2}{d-1}}\bigr{)}, we obtain

[TABLE]

We can now proceed with the proof of the lower variance bounds.

Proof of the lower bounds of Theorem 1.1.

Let $\mathcal{F}$ be the $\sigma$-field defined as above. The conditional variance formula implies that

[TABLE]

As already mentioned, $\mathcal{F}$ induces an independence property. Therefore, we obtain

[TABLE]

Finally, applying Lemma 3.1, Lemma 3.2 and taking expectation yields

[TABLE]

4 Upper variance bounds

In the following, we find upper bounds for $\operatorname{\mathbf{Var}}[V_{\ell}(K_{n})]$, $\ell\in\{1,\dots,d\}$. The proof is based on the Efron-Stein jackknife inequality and follows the ideas of [1]. In contrast to [1], we use the concept of surface body, in particular, Lemma 2.1 about the fact that the surface body is contained in the random polytope $K_{n}$ with high probability. Moreover, we make use of Lemma 2.3 for our estimates.

Proof of the upper bounds of Theorem 1.1.

First, let $K=B^{d}$. We indicate with $T_{n}$ the event that the surface body $K(s\geq\tau_{n})$ is contained in $K_{n}$. Let $\ell\in\{1,\ldots,d\}$. Applying the Efron-Stein jackknife inequality yields

[TABLE]

It is obvious that $(V_{\ell}(K_{n+1})-V_{\ell}(K_{n}))^{2}\leq V_{\ell}(K)^{2}$ and $\operatorname{\mathbf{E}}[\boldsymbol{1}_{T_{n}^{c}}]=\mathbf{P}(T_{n}^{c})$. Since the parameter $\alpha$ can be chosen arbitrarily big in Lemma 2.1, the second term in Equation 6 is negligible in the asymptotic analysis. By Equation 1, we obtain

[TABLE]

If $X_{n+1}|A\in K_{n}|A$, then the set $(K_{n+1}|A)\setminus(K_{n}|A)$ is clearly empty. Otherwise, $(K_{n+1}|A)\setminus(K_{n}|A)$ consists of several disjoint simplices which are the convex hull of $X_{n+1}|A$ and those facets of $K_{n}|A$ that can be “seen” from $X_{n+1}|A$. For $I=\{i_{1},\dots,i_{\ell}\}\subset\{1,\dots,n\}$, we indicate with $F_{I}$ the convex hull of $X_{i_{1}},\dots,X_{i_{\ell}}$. Note that $F_{I}$ and $F_{I}|A$ are $(\ell-1)$-dimensional simplices with probability $1$. The closed half space in $\mathbb{R}^{d}$ which is determined by the hyperplane $A^{\bot}+\operatorname{aff}F_{I}$ and contains the origin is denoted by $H_{0}(F_{I},A)$. The other half space is $H_{+}(F_{I},A)$. The corresponding $\ell$-dimensional half spaces in $A$ are denoted by $H_{0}(F_{I}|A)$ and $H_{+}(F_{I}|A)$. Let $\mathcal{F}(A)$ be the set of $(\ell-1)$-dimensional facets of $K_{n}|A$ that can be seen from $X_{n+1}|A$. It is defined by

[TABLE]

Then,

[TABLE]

Next, the integration is extended over all possible index sets $I,J$ and the order of integration is changed. As a consequence, we obtain

[TABLE]

Note that $[F_{I},X_{n+1}]|A$ and $[F_{J},X_{n+1}]|B$ are contained in the associated caps $C_{\ell}(I,A):=H_{+}(F_{I},A)\cap B^{d}$ and $C_{\ell}(J,B)$. Moreover, we use the abbreviation $C_{d}(I,A)=(H_{+}(F_{I}|A)+A^{\bot})\cap B^{d}$. We indicate with $V_{\ell}(I,A)=\operatorname{vol}_{\ell}(C_{\ell}(I,A))$ and $V_{d}(I,A)=V_{d}(C_{d}(I,A))$ the volumes of these caps. Therefore, the variance is bounded by

[TABLE]

where the summation extends over all $\ell$-tuples $I$ and $J$. Of course, these tuples may have a non-empty intersection. However, if the size of $I\cap J$ is fixed to be $k$, then the corresponding terms in the sum are independent of the choice of $i_{1},\dots,i_{\ell}$ and $j_{1},\dots,j_{\ell}$. For any $k\in\{0,1,\dots,\ell\}$, we indicate with $F$ the convex hull of $X_{1},\dots,X_{\ell}$ and by $G$ the convex hull of $X_{\ell-k+1},\dots,X_{2\ell-k}$. As in [1], we obtain

[TABLE]

We indicate with $\Sigma_{k}$ the $k\text{-th}$ term in the previous sum. By symmetry, we can restrict the summation to those tuples where $V_{d}(I,A)\geq V_{d}(J,B)$. In addition to that, we multiply the integrand by $\boldsymbol{1}\{C_{d}(I,A)\cap C_{d}(J,B)\neq\emptyset\}$. This is indeed possible because the caps have at least the point $X_{n+1}$ in common. It follows immediately that

[TABLE]

Next, we integrate with respect to $x_{2\ell-k+1},\dots,x_{n},x_{n+1}$. Due to the condition $F|A\in\mathcal{F}(A)$, the points $X_{2\ell-k+1},\dots,X_{n}$ are contained in $H_{0}(F,A)$ and $X_{n+1}$ is in $H_{+}(F,A)$. Therefore,

[TABLE]

The assumption $V_{d}(I,A)\geq V_{d}(J,B)$ implies that the height of the cap $C_{d}(I,A)$ is at least the height of $C_{d}(J,B)$. Due to $C_{d}(I,A)\cap C_{d}(J,B)\neq\emptyset$, we find a constant $\beta$ such that $C_{d}(J,B)$ is contained in $\beta\,C_{d}(I,A)$. More precisely, $\beta\,C_{d}(I,A)$ is an enlarged homothetic copy of $C_{d}(I,A)$, where the center of homothety $z\in\SS^{d-1}$ coincides with the center of the cap $C_{d}(I,A)$. It follows from the homogeneity that the Hausdorff measure (restricted to $\beta\,\SS^{d-1}$) of $\beta\,C_{d}(I,A)$ is up to a constant $\mathcal{H}^{d-1}(C_{d}(I,A)\cap\SS^{d-1})$. Therefore,

[TABLE]

As in [1], the conditions $C_{d}(I,A)\cap C_{d}(J,B)\neq\emptyset$ and $V_{d}(I,A)\geq V_{d}(J,B)$ are only satisfied if the angle between $z$ and the subspace $B$ is not larger than twice the central angle $\delta$ of the cap $C_{d}(I,A)$. Moreover, $\delta$ is bounded by

[TABLE]

Thus,

[TABLE]

Due to Lemma 2.3, the condition $T_{n}$ can be replaced by the condition

[TABLE]

for some constant $c_{1}>0$. In the following, the economic cap covering theorem is used, recall Proposition 2.5. Let $h$ be a positive integer such that $2^{-h}\leq\log{n}/n$. Note that the smallest possible value of $h$ is $h_{0}=-\lfloor\log_{2}(\log{n}/n)\rfloor$. According to the economic cap covering theorem, we find for each $h$ a collection of caps $\{C_{1},\dots,C_{m(h)}\}$ which cover the wet part of $B^{d}|A$ with parameter $(2^{-h})^{(\ell+1)/(d-1)}$. This collection of caps is denoted by $\mathcal{M}_{h}$. Each cap $C_{i}$ can be viewed as a projection of a $d$-dimensional cap $C_{i}(A)$ from $B^{d}$ to $A$. Now we consider an arbitrary tuple $(X_{1},\dots,X_{\ell})$ which has a corresponding cap $C_{d}(I,A)$ having volume at most $c_{1}\,(\log{n}/n)^{(d+1)/(d-1)}$. We relate to $(X_{1},\dots,X_{\ell})$ the maximal $h$ such that $C_{\ell}(I,A)\subset C_{i}$ for some $C_{i}\in\mathcal{M}_{h}$. This is indeed possible since at least $2^{-h_{0}}$ is roughly $\log{n}/n$ and the volume of the caps in $\mathcal{M}_{h}$ tends to zero as $h\to\infty$. As a consequence, we obtain

[TABLE]

and

[TABLE]

According to Lemma 2.3, $\mathcal{H}^{d-1}(C_{d}(I,A)\cap\SS^{d-1})\leq\mathcal{H}^{d-1}(C_{i}(A)\cap\SS^{d-1})\ll 2^{-h}$. Due to the maximality of $h$, it holds $V_{d}(I,A)\geq 2^{-(h+1)(d+1)/(d-1)}$. In addition to that, it follows from Lemma 2.3 that $\mathcal{H}^{d-1}(C_{d}(I,A)\cap\SS^{d-1})\geq c_{2}2^{-(h+1)}$, for some constant $c_{2}>0$. Therefore, we obtain

[TABLE]

Then, we integrate each $(X_{1},\dots,X_{\ell})$ on $(C_{i}(A))^{\ell}$ and we use the fact $1-x\leq\exp(-x)$ to obtain

[TABLE]

Since the volume of the wet part of $B^{\ell}$ with parameter $2^{-h(\ell+1)/(d-1)}$ is \Theta\bigl{(}2^{-2h/(d-1)}\bigr{)} (note that $h\to\infty$, as $n\to\infty$), we obtain

[TABLE]

Finally, this results in

[TABLE]

Note that we used Lemma 2.6 and Equation 11 in the last step. As in [1], we divide the previous sum into two parts in order to see the magnitude of the variance. The integer $h_{1}$ is defined by

[TABLE]

On the one hand, we have

[TABLE]

On the other hand, let $i=h_{1}-h$. Then, we can perform the following estimate, namely,

[TABLE]

As a consequence, it holds

[TABLE]

Finally, the upper bounds are proven by summing up all $\Sigma_{k}$, $k=0,\dots,\ell$, in Equation 9.

In order to extend the proof to the case of a convex body $K\in\mathcal{K}^{2}_{+}$, we follow the ideas presented in [1, Section 6]. By the compactness of $\partial K$, there exist $\gamma>0$ and $\Gamma>0$, the global upper and the global lower bound on the principal curvatures of $\partial K$, respectively. In our setting, all projected images of $\partial K$ also have a boundary with the same properties as $\partial K$, see for example [13, Remark 5]. Without loss of generality we can choose $\gamma$ and $\Gamma$ to be also a bound on the principal curvatures of the boundaries of all $\ell\text{-dimensional}$ projections of $K$. Hence, one can locally approximate $\partial K$ with affine images of balls and the volume of a $\ell\text{-dimensional}$ cap with height $t>0$ has order $t^{\frac{\ell+1}{2}}$. Finally, [1, Equation (27)] ensures that Equation 10 still holds. ∎

In the fashion of [1, Section 7], we derive strong laws of large numbers from the upper variance bounds together with the following result of [8].

Proposition 4.1**.**

[8, Theorem 1]* Let $K\in\mathcal{K}^{2}_{+}$ and choose $n$ random points on $\partial K$ independently and according to the probability distribution $\mathcal{H}^{d-1}$. Then, there exist positive constants $c_{d,\ell,K}$ depending on $d,\ell$ and the principal curvatures of $K$ such that*

[TABLE]

For the sake of brevity, the explicit expression of $c_{d,\ell,K}$ is omitted here. It can be found in [8, Equation (2)].

Proof of Theorem 1.2.

Let $\ell\in\{1,\dots,d\}$. Chebyshev’s inequality and the variance upper bound yield

[TABLE]

Select now the subsequence of indices $n_{k}=k^{2}$. Then, it follows

[TABLE]

Applying the Borel-Cantelli Lemma together with Equation 12, we obtain that

[TABLE]

holds with probability 1. Note that $V_{\ell}(K)-V_{\ell}(K_{n})$ is a decreasing and positive sequence. Therefore, this gives

[TABLE]

whenever $n_{k-1}\leq n\leq n_{k}$. Taking the limit as $k\to\infty$, $n_{k-1}/n_{k}\to 1$, which allows us to conclude that the desired limit is reached by the whole sequence with probability $1$. ∎

5 Central limit theorems

In this last section, we prove the central limit theorems. In contrast to [16], where floating bodies were used, here we work with surface bodies as it was already done in [15] for the case of the volume. In addition to that, we make use of the normal approximation bound of Proposition 2.7.

Proof of Theorem 1.3.

First, we prove the central limit theorems for $K=B^{d}$. For this reason, let us introduce the two events $B_{1}$ and $B_{2}$. The event that the random polytope $[X_{2},\dots,X_{n}]$ contains the surface body $K(s\geq\tau_{n})$ is denoted by $B_{1}$. Due to the definition of $B_{1}$, it follows by Lemma 2.1 that

[TABLE]

where $c_{1}\in(0,\infty)$ is independent of $n$. We denote by $B_{2}$ the event that the random polytope $\bigcap_{W\in\{Y,Y^{\prime},Z,Z^{\prime}\}}[W_{4},\ldots,W_{n}]$ contains the surface body $K(s\geq\tau_{n})$, where $Y,Y^{\prime},Z,Z^{\prime}$ are recombinations of the random vector $X=(X_{1},\ldots,X_{n})$. By taking the union bound, we obtain

[TABLE]

where $c_{2}\in(0,\infty)$ is again independent of $n$. Next, for any $\ell\in\{1,\ldots,d\}$, we apply the bound in Proposition 2.7 to the random variables

[TABLE]

Note that $D_{i}W=D_{i}V_{\ell}(K_{n})$ and $D_{i_{1},i_{2}}W=D_{i_{1},i_{2}}V_{\ell}(K_{n})$ for $i,i_{1},i_{2}\in\{1,\ldots,n\}$. Conditioned on the event $B_{1}$, we obtain from (1),

[TABLE]

We now define a full-dimensional cap $C$ in such a way that $K_{n}\setminus[X_{2},\ldots,X_{n}]$ is contained in $C$. Consider now the visibility region $\mathrm{Vis}_{X_{1}}(\tau_{n})$ of $X_{1}$. By definition of the event $B_{1}$, the surface body and by Lemma 2.3, the diameter of this visibility region is at most $c_{3}\tau_{n}^{1/(d-1)}$, where $c_{3}>0$. We now indicate with $D(X_{1},c_{3}\tau_{n}^{1/(d-1)})$ the points on $\partial K$ with distance at most $c_{3}\tau_{n}^{1/(d-1)}$ from $X_{1}$. Then, C\coloneqq\operatorname{conv}\bigl{\{}D(X_{1},c_{3}\tau_{n}^{1/(d-1)})\bigr{\}} is a spherical cap and it follows from Lemma 2.4 that $C$ has volume of order at most $\tau_{n}^{(d+1)/(d-1)}$. We call $\alpha$ the central angle of $C$. For any subspace $L\in G(d,\ell)$, it holds that $(K_{n}|L)\setminus([X_{2},\ldots,X_{n}]|L)\subseteq(C|L)$. We obtain $\operatorname{vol}_{\ell}(C|L)\ll\tau_{n}^{(\ell+1)/(d-1)}$. Indeed, the height of $C|L$ has the same order as the height of $C$, namely $\tau_{n}^{2/(d-1)}$, while the order of its base changes from $((\tau_{n})^{1/(d-1)})^{d-1}$ to $((\tau_{n})^{1/(d-1)})^{\ell-1}$, since the dimension of $L$ is $\ell$. By construction of $C$, it now follows that if $\sphericalangle(X_{1},L)$, the angle between $X_{1}$ and $L$, is too wide compared to $\alpha$, then $C|L\subseteq K_{n}|L$, for sufficiently large $n$. Whenever this occurs, it also holds in particular that $(K_{n}\setminus[X_{2},\ldots,X_{n}])|L\subseteq K_{n}|L$, i.e., $K_{n}|L=[X_{2},\ldots,X_{n}]|L$. In fact, one can check that the integrand in (13) can only be non-zero if $\sphericalangle(X_{1},L)\ll\alpha$. Therefore, we can restrict the integration to the set $\{L\in G(d,\ell):\sphericalangle(X_{1},L)\ll\alpha\}$. Moreover, it holds that $\alpha\ll V_{d}(C)^{1/(d+1)}$, see e.g. [1, Equation (21)]. According to Lemma 2.6, this gives

[TABLE]

Putting everything together, we see that

[TABLE]

On the complement $B_{1}^{c}$ of $B_{1}$ we use the trivial estimate $D_{1}V_{\ell}(K_{n})\leq V_{\ell}(K)$. Since $\mathbf{P}(B_{1}^{c})\ll n^{-\alpha}$, we obtain

[TABLE]

for all $p\geq 1$. As a consequence, we can bound the terms in the normal approximation bound which involve $\gamma_{3}$ and $\gamma_{4}$. Thus,

[TABLE]

By using the Cauchy-Schwarz inequality, we can estimate $\gamma_{5}$ as well. Namely,

[TABLE]

Thus, we obtain

[TABLE]

In the next step, we consider the terms involving the second difference operator. On the event $B_{2}$ it may be concluded from (14) that $D_{i}f(V)^{2}\ll(\log{n}/n)^{2{d+1\over d-1}}$ for all $i\in\{1,2,3\}$ and $V\in\{Z,Z^{\prime}\}$. Moreover, we note that on $B_{2}$ the following inclusions hold

[TABLE]

The same applies to $D_{1,3}f(Y^{\prime})$. Thus,

[TABLE]

We note that the diameter of the previous union is at most $c_{4}\tau_{n}^{1/(d-1)}$, where $c_{4}>0$. As before, we define the spherical cap $C^{\prime}\coloneqq\operatorname{conv}\{D(z,c_{4}\tau_{n}^{1/(d-1)})\}$. It follows from Lemma 2.4 that $C^{\prime}$ has volume of order at most $\tau_{n}^{(d+1)/(d-1)}$. We obtain

[TABLE]

where for the last inequality we have used Lemma 2.3. On the event $B_{2}^{c}$ we use the trivial estimate $V_{\ell}(K)$ for all difference operators and estimate all indicators by one. Since $\mathbf{P}(B_{2}^{c})\ll n^{-\alpha}$, we obtain

[TABLE]

Analogously, we can bound $\gamma_{1}$. Indeed, suppose that $Y_{1}=Y^{\prime}_{1}$ (by independence, $Y_{1}\neq Y^{\prime}_{1}$ gives a smaller order), then

[TABLE]

and we obtain

[TABLE]

Thus,

[TABLE]

Finally,

[TABLE]

Considering all the estimates together, we obtain by Proposition 2.7

[TABLE]

For the case of a generic $K\in\mathcal{K}^{2}_{+}$ we argue as at the end of the proof of the upper bounds of Theorem 1.1. Because of the global bounds on the principal curvatures and the local approximation of $\partial K$ with affine images of balls, the construction of $C$ and the relations regarding its volume, its central angle and the subspaces $L$ which ensure $C|L\subseteq K_{n}|L$ are not afflicted. In particular, the asymptotic bounds $\operatorname{vol}_{\ell}(C|L)\ll\tau_{n}^{{(\ell+1)}/{(d-1)}}$, $\alpha\ll V_{d}(C)^{1/(d+1)}\ll\tau_{n}^{1/(d-1)}$ and $\sphericalangle(X_{1},L)\ll\alpha$ stated above still hold, with the difference that the implicit constants depend on $\gamma$ and $\Gamma$, the bounds on the principal curvatures of $\partial K$. Then, the proof can be completed like in the case of the ball. ∎

Acknowledgement

We would like to thank Christoph Thäle for helpful discussions and insights.