Approximations of convex bodies by measure-generated sets

TL;DR

This paper introduces measure-generated convex sets and explores their properties to improve approximation of convex bodies by polytopes, extending the vertex index and providing bounds for centroid bodies of probability measures.

Contribution

It defines and analyzes measure-generated convex sets and applies these to enhance convex body approximation and extend the vertex index concept.

Findings

Properties of measure-generated convex sets are characterized.

New bounds are established for approximation of convex bodies.

An extension of the vertex index is proposed and analyzed.

Abstract

Given a Borel measure $\mu$ on ${\mathbb R}^{n}$, we define a convex set by \[ M({\mu})=\bigcup_{\substack{0\le f\le1,\\ \int_{{\mathbb R}^{n}}f\,{\rm d}{\mu}=1 } }\left\{ \int_{{\mathbb R}^{n}}yf\left(y\right)\,{\rm d}{\mu}\left(y\right)\right\} , \] where the union is taken over all $\mu$-measurable functions $f:{\mathbb R}^{n}\to\left[0,1\right]$ with $\int_{{\mathbb R}^{n}}f\,{\rm d}{\mu}=1$. We study the properties of these measure-generated sets, and use them to investigate natural variations of problems of approximation of general convex bodies by polytopes with as few vertices as possible. In particular, we study an extension of the vertex index which was introduced by Bezdek and Litvak. As an application, we provide a lower bound for certain average norms of centroid bodies of non-degenerate probability measures.

Equations

$${\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=\bigcup_{\begin{subarray}{c}0\leq f\leq 1,\\ \int_{\mathbb{R}^{n}}f\,{\rm d}\mu=1\end{subarray}}\mathopen{}\mathclose{{}\left\{\int_{\mathbb{R}^{n}}yf\mathopen{}\mathclose{{}\left(y}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(y}\right)}\right\},$$

$$d_{R}\mathopen{}\mathclose{{}\left(K}\right)=\inf\mathopen{}\mathclose{{}\left\{N\in\mathbb{N}\,:\,\exists\,P={\rm conv}\mathopen{}\mathclose{{}\left(x_{1},\dots,x_{N}}\right)\subseteq\mathbb{R}^{n}\,,\,\frac{1}{R}P\subseteq K\subseteq P}\right\},$$

$$d_{R}\mathopen{}\mathclose{{}\left(K}\right)\leq e^{cn\log R/R^{2}}$$

$$d_{R}\mathopen{}\mathclose{{}\left(K}\right)\leq ne^{cn/R}.$$

$$D_{R}\mathopen{}\mathclose{{}\left(K}\right)=\inf\mathopen{}\mathclose{{}\left\{\sum_{i=1}^{N}\mathopen{}\mathclose{{}\left\|x_{i}}\right\|_{K}\,:\,\exists\,P={\rm conv}\mathopen{}\mathclose{{}\left(x_{1},\dots,x_{N}}\right)\subseteq\mathbb{R}^{n}\,,\,\frac{1}{R}P\subseteq K\subseteq P}\right\}.$$

$$2n\leq{\rm vein}\mathopen{}\mathclose{{}\left(K}\right)\leq 24n^{3/2}$$

$${\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=\bigcup_{\begin{subarray}{c}0\leq f\leq 1,\\ \int_{\mathbb{R}^{n}}f\,{\rm d}\mu=1\end{subarray}}\mathopen{}\mathclose{{}\left\{\int_{\mathbb{R}^{n}}yf\mathopen{}\mathclose{{}\left(y}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(y}\right)}\right\},$$

$${\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{n}\,:\,\exists\mathopen{}\mathclose{{}\left\{\lambda_{i}}\right\}_{i=1}^{N}\,\text{s.t.}\,0\leq\lambda_{i}\leq w_{i},\,\sum_{i=1}^{N}\lambda_{i}=1,\,\text{and }x=\sum_{i=1}^{N}\lambda_{i}x_{i}}\right\}.$$

$$d_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right)=\inf\mathopen{}\mathclose{{}\left\{\mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right)\,:\,\frac{1}{R}{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)}\right\},$$

$$D_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right)=\inf\mathopen{}\mathclose{{}\left\{\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)\,:\,\frac{1}{R}{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)}\right\}.$$

$${\rm vein^{*}}\mathopen{}\mathclose{{}\left(K}\right)=\inf\mathopen{}\mathclose{{}\left\{\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}d\mu\mathopen{}\mathclose{{}\left(x}\right)\,:\,K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)}\right\}.$$

$$d_{\sqrt{n}}^{*}\mathopen{}\mathclose{{}\left(K}\right)\leq C,\,\,\text{and}\,\,\,D_{\sqrt{n}}^{*}\mathopen{}\mathclose{{}\left(K}\right)\leq Cn.$$

$$d_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right)\leq\exp\mathopen{}\mathclose{{}\left(1+\frac{n-1}{R-1}}\right),\,\,\text{and}\,\,D_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right)\leq R\exp\mathopen{}\mathclose{{}\left(1+\frac{n-1}{R-1}}\right).$$

$${\rm vein^{*}}\mathopen{}\mathclose{{}\left(K}\right)\geq c\sqrt{n}.$$

$$h_{Z_{p}\mathopen{}\mathclose{{}\left(\mu}\right)}\mathopen{}\mathclose{{}\left(\theta}\right)=\mathopen{}\mathclose{{}\left(\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left|\langle x,\,\theta\rangle}\right|^{p}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)}\right)^{1/p},$$

$$\mathopen{}\mathclose{{}\left(\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{Z_{p}\mathopen{}\mathclose{{}\left(\mu}\right)}^{p}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)}\right)^{1/p}\apprge\frac{\sqrt{n}}{p}.$$

$$\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{Z_{1}\mathopen{}\mathclose{{}\left(\mu}\right)}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)\geq c\sqrt{n}.$$

$$R\mathopen{}\mathclose{{}\left(\theta}\right):=\max\{R\in\mathbb{R}\,,\,\mu\mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left(x\in\mathbb{R}^{n}\,:\,\langle x,\,\theta\rangle\geq R\}}\right)\geq 1}\right\}.$$

$$\displaystyle f_{\theta}(t)=\mathopen{}\mathclose{{}\left\{\begin{array}[]{cc}0,&\text{ if $t<R$,}\\ 1,&\text{ if $t>R$,}\\ 0,&\text{ $t=R$ and $\mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{\langle x,\,\theta\rangle=R\mathopen{}\mathclose{{}\left(\theta}\right)}\right\}}\right)=0$,}\\ \frac{1-\mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{\langle x,\,\theta\rangle>R(\theta)}\right\}}\right)}{\mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{\langle x,\theta\rangle=R(\theta)}\right\}}\right)},&\text{ $t=R$ and $\mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{\langle x,\,\theta\rangle=R\mathopen{}\mathclose{{}\left(\theta}\right)}\right\}}\right)\neq 0$}.\end{array}}\right.$$

$$y_{\theta}:=\int_{\mathbb{R}^{n}}xf_{\theta}\mathopen{}\mathclose{{}\left(\langle x,\,\theta\rangle}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right).$$

$$⟨y_{\theta },\theta ⟩-⟨y,\theta ⟩$$

$$\langle y_{\theta},\,\theta\rangle-\langle y,\,\theta\rangle\geq\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left(f_{\theta}\mathopen{}\mathclose{{}\left(\langle x,\,\theta\rangle}\right)-f(x)}\right)R\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=0.$$

$$h_{K}\mathopen{}\mathclose{{}\left(\theta}\right)\leq\int_{H_{\theta}^{+}}\langle x,\,\theta\rangle\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right).$$

$$\displaystyle h_{K}\mathopen{}\mathclose{{}\left(\theta}\right)$$

$$Z\mathopen{}\mathclose{{}\left(x_{1},\dots,x_{m}}\right)=\mathopen{}\mathclose{{}\left\{\lambda_{1}x_{1}+\dots+\lambda_{m}x_{m}\,:\,\lambda_{1},\dots,\lambda_{m}\in\mathopen{}\mathclose{{}\left[0,1}\right]}\right\}.$$

$${\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq{\rm conv}\mathopen{}\mathclose{{}\left(x_{1},\cdots,x_{m}}\right)\cap Z\mathopen{}\mathclose{{}\left(w_{1}x_{1},\dots,w_{m}x_{m}}\right).$$

$${\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq P:=\mathopen{}\mathclose{{}\left\{\sum_{i=1}^{m}\lambda_{i}w_{i}x_{i}\,:\>\lambda_{i}\geq 0,\,\sum_{i=1}^{m}\lambda_{i}w_{i}=1}\right\}={\rm conv}\mathopen{}\mathclose{{}\left(x_{1},\dots,x_{m}}\right).$$

$${\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq Z:=Z\mathopen{}\mathclose{{}\left(w_{1}x_{1},\dots w_{m}x_{m}}\right)=\mathopen{}\mathclose{{}\left\{\sum_{i=1}^{m}\lambda_{i}w_{i}x_{i}\,:\,0\leq\lambda_{i}\leq 1}\right\}.$$

$$Z\mathopen{}\mathclose{{}\left(\mu}\right)=\mathopen{}\mathclose{{}\left\{\int_{\mathbb{R}^{n}}xf\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)\,:\,0\leq f\leq 1}\right\},\,P\mathopen{}\mathclose{{}\left(\mu}\right)=\mathopen{}\mathclose{{}\left\{\int_{\mathbb{R}^{n}}xf\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)\,:\,0\leq f,\,\int_{\mathbb{R}^{n}}f\,{\rm d}\mu=1}\right\}.$$

$$h_{{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)}(\theta)=\frac{1}{2}\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left|\langle\theta,\,x\rangle}\right|\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right).$$

Full text

[TABLE]

where the union is taken over all $\mu$-measurable functions f:\mathbb{R}^{n}\to\mathopen{}\mathclose{{}\left[0,1}\right] with $\int_{\mathbb{R}^{n}}f\,{\rm d}\mu=1$. We study the properties of these measure-generated sets, and use them to investigate natural variations of problems of approximation of general convex bodies by polytopes with as few vertices as possible. In particular, we study an extension of the vertex index which was introduced by Bezdek and Litvak. As an application, we provide a lower bound for certain average norms of centroid bodies of non-degenerate probability measures.

1. Introduction

1.1. Background and motivation

Problems pertaining to approximation, on their various aspects and applications, have been extensively studied in the theory of convex bodies, see e.g. [12], and [9].

An example for such a problem is that of approximating a convex body, namely a compact convex set with non-empty interior, by a polytope (the convex hull of finitely many points) with as few vertices as possible, within a given Banach-Mazur distance. More precisely, for any convex body $K\subseteq\mathbb{R}^{n}$ centered at the origin, and $R>1$ we define:

[TABLE]

where {\rm conv}\mathopen{}\mathclose{{}\left(x_{1},\dots,x_{N}}\right) is the convex hull of $x_{1},\dots,x_{N}$.

A result of Barvinok [3] implicitly states that for any centrally-symmetric convex body $K\subseteq\mathbb{R}^{n}$ and $2<R<\sqrt{n}$,

[TABLE]

for some universal constant $c>0$. In particular, d_{c\sqrt{n}}\mathopen{}\mathclose{{}\left(K}\right)\leq n. We also mention the result of Szarek [27] who shows that for any convex body with center of mass at the origin and $2<R<n$,

[TABLE]

For the case $R=n$, a similar result to that of Szarek can be found in [8].

We remark that both approaches in [3] and [4] work in the fine scale regime, for which an optimal result was very recently proven in [22].

It is also worth pointing out that there is still a large gap between the symmetric and the non-symmetric case. For example, it is not clear whether in the non-symmetric case d_{\sqrt{n}}\mathopen{}\mathclose{{}\left(K}\right) can have a polynomial bound in $n$.

Note that for the special case $R=\infty$, d_{\infty}\mathopen{}\mathclose{{}\left(K}\right) trivially equals $n+1$, e.g., by scaling away the vertices of a centered simplex. However, replacing the number of vertices of the approximating polytope by a different “cost” leads to the following quantity:

[TABLE]

Here, \mathopen{}\mathclose{{}\left\|\cdot}\right\|_{K} stands for the gauge function of $K$ which, in the case where $K=-K$, is the norm on $\mathbb{R}^{n}$ which is induced by $K$. This quantity is also linear-invariant, and is equivalent to d_{R}\mathopen{}\mathclose{{}\left(K}\right) for any finite $R$ in the sense that d_{R}\mathopen{}\mathclose{{}\left(K}\right)\leq D_{R}\mathopen{}\mathclose{{}\left(K}\right)\leq Rd_{R}\mathopen{}\mathclose{{}\left(K}\right). However, D_{\infty}\mathopen{}\mathclose{{}\left(K}\right) is no longer trivial. In fact, it coincides with the vertex index of $K$, denoted by {\rm vein}\mathopen{}\mathclose{{}\left(K}\right), which was introduced by Bezdek and Litvak in [5], and further studied in [10] and [11]. For example, it was shown that for any centrally-symmetric convex body $K\subseteq\mathbb{R}^{n}$,

[TABLE]

The lower bound, which is attained for $B_{1}^{n}$, was proved in [10], and the upper bound, which (up to a universal constant) is attained for the Euclidean unit ball $B_{2}^{n}$, was proved in [11]. We remark that the choice of the $l_{1}$ cost \sum_{i=1}^{N}\mathopen{}\mathclose{{}\left\|x_{i}}\right\|_{K} seems arbitrary and can be replaced by different linear-invariant costs, such as \mathopen{}\mathclose{{}\left(\sum\mathopen{}\mathclose{{}\left\|x_{i}}\right\|_{K}^{p}}\right)^{1/p} for any $p\geq 1$.

1.2. Metronoids

The main purpose of this note is to introduce a natural way of generating convex bodies from Borel measures, along with associated costs, and study new quantities which are closely related to D_{R}\mathopen{}\mathclose{{}\left(K}\right), d_{R}\mathopen{}\mathclose{{}\left(K}\right), and {\rm vein}\mathopen{}\mathclose{{}\left(K}\right). Our construction goes as follows:

Definition 1.1**.**

Given a Borel measure $\mu$ on $\mathbb{R}^{n}$, we define

[TABLE]

where the union is taken over all measurable functions f:\mathbb{R}^{n}\to\mathopen{}\mathclose{{}\left[0,1}\right] with $\int_{\mathbb{R}^{n}}f\,{\rm d}\mu=1$. We call the set {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq\mathbb{R}^{n}, the metronoid111originating from the greek word “metron” for “measure” (the authors thank B. Vritsiou for the greek lesson). generated by $\mu$

Note that {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) is always a closed convex set, which is bounded if $\mu$ has finite first moment. In particular, for $x_{1},\dots,x_{N}\in\mathbb{R}^{n}$ the discrete measure $\mu=\sum_{i=}^{N}\delta_{x_{i}}$ generates the convex hull of \mathopen{}\mathclose{{}\left\{x_{1},\dots,x_{N}}\right\}. By adding weights, $w_{1},\dots,w_{N}>0$, that is, considering the weighted measure $\mu=\sum_{i=1}^{N}w_{i}\delta_{x_{i}}$, the generated convex body {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) becomes a “weighted convex hull”, where each point $x_{i}$ can only participate in the convex hull with a coefficient $\lambda_{i}$ whose maximal value is $w_{i}$ . In other words, we have:

[TABLE]

Also note that if \mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right)<1 then {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=\emptyset, and if \mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right)=1 then {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) is the singleton \mathopen{}\mathclose{{}\left\{\int x\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)}\right\}, namely the center of mass of $\mu$.

One may consider various other interesting classes of metronoids, for example, the class of bodies generated by uniform measures on convex bodies, which turn out to be closely related to floating bodies. For detailed discussion on special classes of metronoids and their properties, see Section 2 below.

The notion of metronoids leads to the following variations of d_{R}\mathopen{}\mathclose{{}\left(K}\right) and D_{R}\mathopen{}\mathclose{{}\left(K}\right):

[TABLE]

and

[TABLE]

Clearly, we have that d_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right)\leq d_{R}\mathopen{}\mathclose{{}\left(K}\right), and D_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right)\leq D_{R}\mathopen{}\mathclose{{}\left(K}\right). One can also verify that the above quantities are both linear-invariant. While it is plausible that the family of metronoids generated by all finite Borel measure coincides with the family of all convex bodies, it is still interesting to consider the approximation by metronoids since for different values of $R$, the associated costs \mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right) and \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\mu are not necessarily minimized for $\mu$ for which {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=K.

For $R=\infty$, we obtain the following variation of the vertex index, which we refer to as the fractional vertex index:

[TABLE]

We remark that the motivation of Bezdek and Litvak to study the vertex index is its relation to Hadwiger’s famous problem of illuminating a convex body by light sources, and to the Gohberg-Markus-Hadwiger equivalent problem of covering a convex body by smaller copies of itself (see e.g., [6] and references therein). Fractional versions of the illumination and covering problems were studied in [21] and [2].

1.3. Main results

1.3.1. Upper and lower bounds. Our first main result provides a

bound for d_{\sqrt{n}}^{*}\mathopen{}\mathclose{{}\left(K}\right) and D_{\sqrt{n}}^{*}\mathopen{}\mathclose{{}\left(K}\right) in the centrally-symmetric case:

Theorem 1.2**.**

There exists a universal constant $C>0$ such that for every centrally-symmetric convex body $K\subseteq\mathbb{R}^{n}$, one has

[TABLE]

Our second main result provides a general upper for d_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right) and D_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right):

Theorem 1.3**.**

Let $K\subseteq\mathbb{R}^{n}$ be a centered convex body. Then for $1<R\leq n$ one has

[TABLE]

Note that Theorems 1.2 and 1.3 are reminiscent of 1.1 and 1.2, but do not follow from them formally. We believe that a further investigation of d_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right) and D_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right) in the non-symmetric case may shed light on the classical counterpart d_{R}\mathopen{}\mathclose{{}\left(K}\right) and D_{R}\mathopen{}\mathclose{{}\left(K}\right), e.g., in the regime of $R\approx\sqrt{n}$.

An immediate corollary of Theorem 1.3 provides the following upper bound for the fractional vertex index,

Corollary 1.4**.**

For every centered convex body $K\subseteq\mathbb{R}^{n}$ one has {\rm vein^{*}}\mathopen{}\mathclose{{}\left(K}\right)\leq D_{n}^{*}\mathopen{}\mathclose{{}\left(K}\right)\leq e^{2}n.

Our third main result provides a lower bound for the fractional vertex index in the centrally-symmetric case

Theorem 1.5**.**

There exists a universal constant $c>0$ such that for every centrally-symmetric convex body $K\subseteq\mathbb{R}^{n}$, one has:

[TABLE]

We remark that, up to a constant, Corollary 1.4 is sharp for the cross-polytope B_{1}^{n}=\mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left(x_{1},\dots,x_{n}}\right)^{T}\in\mathbb{R}^{n}\,:\,\sum_{i=1}^{n}\mathopen{}\mathclose{{}\left|x_{i}}\right|\leq 1}\right\}, and Theorem 1.5 is sharp for the Euclidean unit ball B_{2}^{n}=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{n}\,:\,\mathopen{}\mathclose{{}\left|x}\right|^{2}\leq 1}\right\}. In fact, in Section 4.1 we show that {\rm vein^{*}}\mathopen{}\mathclose{{}\left(B_{1}^{n}}\right)=2n , and {\rm vein^{*}}\mathopen{}\mathclose{{}\left(B_{2}^{n}}\right)=\sqrt{2\pi n}\mathopen{}\mathclose{{}\left(1+{\rm o}\mathopen{}\mathclose{{}\left(1}\right)}\right).

The proof of Theorem 1.5 employs a proportional Dvoretzky-Rogers factorization Theorem by Bourgain and Szarek [14]. However, we suspect that a proof by symmetrization should show that the extremizer in the general case is $B_{2}^{n}$:

Conjecture 1.6**.**

For any centered convex body $K\subseteq\mathbb{R}^{n}$, {\rm vein^{*}}\mathopen{}\mathclose{{}\left(K}\right)\geq{\rm vein^{*}}\mathopen{}\mathclose{{}\left(B_{2}^{n}}\right)\approx\sqrt{n}.

1.3.2. An application to centroid bodies.

The $L_{p}$-centroid bodies were introduced by Lutwak and Zhang [19] (under different normalization than we use below) and have been studied extensively by various authors. In particular, $L_{p}$-centoid bodies have become an indispensable part of the theory of asymptotic convex geometry since the seminal work of Paouris [23]. For a survey on this subject, see [7, Ch. 5], and references therein.

Given $p\geq 1$ and a Borel probability measure $\mu$ with bounded $p^{th}$ moment, the $L_{p}$-centroid body Z_{p}\mathopen{}\mathclose{{}\left(\mu}\right) is defined by the relation

[TABLE]

where $\langle\cdot,\,\cdot\rangle$ stands for the standard Euclidean inner product on $\mathbb{R}^{n}$, and h_{K}\mathopen{}\mathclose{{}\left(\theta}\right)=\sup_{K}\langle x,\,\theta\rangle is the support function of a convex body $K\subseteq\mathbb{R}^{n}$ (see e.g., [25] for properties of supporting functionals).

For a log-concave measure $\mu$, the bodies Z_{p}\mathopen{}\mathclose{{}\left(\mu}\right) admit many remarkable properties due to the phenomenon of concentration of measure. For example, reverse Hölder inequalities for norms, which imply that, for some universal constant $c>0$, Z_{p}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq Z_{q}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq c\frac{q}{p}\,Z_{p}\mathopen{}\mathclose{{}\left(\mu}\right) for any $1\leq p\leq q$. Moreover, for $p\geq 1$, one has

[TABLE]

It turns out that for $p=1$, the above estimation holds without the assumption that $\mu$ is log-concave. In fact, this result is a direct corollary of Theorem 1.5:

Corollary 1.7**.**

There exists a universal constant $c>0$ such that for any non-degenerate probability Borel measure $\mu$ with bounded first moment, one has

[TABLE]

We remark that the proof of Corollary 1.7 (or, equivalently, of Theorem 1.5) is based on high-dimensional phenomena, rather than concentration of measure (which is used to obtain the same result in the case of log-concave measures). Other results in the spirit of Corollary 1.7, where the log-concavity assumption on the measure may be relaxed, can be found in [15, 24, 16, 17, 18].

This paper is organized as follows. In Section 2 we study the properties of metronoids, including a general characterization of their support functions, descriptions of several classes of metronoids, and the various properties of metronoids generated by discrete measures. In Section 3, we prove Theorems 1.2 and 1.3. In Section 4, we discuss the fractional vertex index, provide precise computations of the fractional vertex index of $B_{1}^{n}$ and $B_{2}^{n}$, and prove Theorem 1.5. We conclude this paper with a proof of Corollary 1.7 in Section 5.

Acknowledgements. The authors thank Alon Nishry and Beatrice Vritsiou for useful discussions. The second named author thanks Shiri Artstein-Avidan for helpful conversations on possible extensions of the vertex index, and for her comments regarding the written text.

2. Properties of Metronoids

2.1. Descriptions of Metronoids

In this section we give several geometric descriptions of metronoids in terms of their generating measures.

2.1.1. A general characterization

Let $\mu$ be any finite Borel measure on $\mathbb{R}^{n}$. We begin with providing a formula for the support function of {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right).

For each $\theta\in\mathbb{S}^{n-1}$, define

[TABLE]

Correspondingly, we define $f_{\theta}:\mathbb{R}\rightarrow[0,1]$ as follows:

[TABLE]

One can easily verify that $0\leq f_{\theta}\leq 1$, and \int_{\mathbb{R}^{n}}f_{\theta}\mathopen{}\mathclose{{}\left(\langle x,\,\theta\rangle}\right))\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=1. Therefore,

[TABLE]

The following proposition describes the support function of {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) in direction $\theta$, in terms of $y_{\theta}$:

Proposition 2.1**.**

With the notation above, for any y\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right), and $\theta\in\mathbb{S}^{n-1}$, $\langle y,\theta\rangle\leq\langle y_{\theta},\theta\rangle$. Namely, h_{{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)}(\theta)=\langle y_{\theta},\theta\rangle.

Proof.

Fix $\theta\in\mathbb{S}^{n-1}$ and let y\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). Then there exists a function $0\leq f(x)\leq 1$ such that \int_{\mathbb{R}^{n}}f(x)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=1 and \int_{\mathbb{R}^{n}}xf(x)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=y. Then, denoting R=R\mathopen{}\mathclose{{}\left(\theta}\right), we have:

[TABLE]

By the definition of $f_{\theta},$ it follows that f_{\theta}\mathopen{}\mathclose{{}\left(\langle x,\,\theta\rangle}\right)-f(x)\geq 0 whenever $\langle x,\theta\rangle>R$, and f_{\theta}\mathopen{}\mathclose{{}\left(\langle x,\,\theta\rangle}\right)-f(x)\leq 0 whenever $\langle x,\theta\rangle<R$. Therefore, we have that for every $x\in\mathbb{R}^{n}$, \mathopen{}\mathclose{{}\left(f_{\theta}\mathopen{}\mathclose{{}\left(\langle x,\,\theta\rangle}\right)-f(x)}\right)\langle x,\,\theta\rangle\geq\mathopen{}\mathclose{{}\left(f_{\theta}\mathopen{}\mathclose{{}\left(\langle x,\,\theta\rangle}\right)-f(x)}\right)R, which together which the above equality implies that

[TABLE]

∎

For each $\theta\in\mathbb{S}^{n-1}$, define H_{\theta}^{+}:=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{n}\,:\,\langle x,\,\theta\rangle>0}\right\}. In the sequel, we will also need the following useful fact:

Proposition 2.2**.**

Let $K\subseteq\mathbb{R}^{n}$ be a convex body. Suppose $\mu$ is a measure such that K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). Then for every $\theta\in\mathbb{S}^{n-1}$ we have that

[TABLE]

Proof.

Fix $\theta\in\mathbb{S}^{n-1}$, and let $x_{\theta}\in K$ such that h_{K}\mathopen{}\mathclose{{}\left(\theta}\right)=\langle x_{\theta},\,\theta\rangle. Since K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right), there exists a function $0\leq f\leq 1$ such that x_{\theta}=\int_{\mathbb{R}^{n}}xf\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right), and hence

[TABLE]

∎

2.1.2. Discrete measures

In this section we provide some geometric description of metronoids that are generated by discrete measures.

The first property states that the metronoid generated by a finite discrete measure is a polytope:

Proposition 2.3**.**

Let $x_{1},\dots,x_{m}\in\mathbb{R}^{n}$, $w_{1},\dots,w_{m}>0$, and define $\mu=\sum_{i=1}^{m}w_{i}\delta_{x_{i}}$. Then {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) is a polytope.

Proof.

Consider the linear map $F:\mathbb{R}^{m}\to\mathbb{R}^{n}$ defined by F\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\lambda_{1},\dots,\lambda_{m}}\right)}\right)=\sum_{i=1}^{m}\lambda_{i}w_{i}x_{i}, and consider the polytope P=\mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left(\lambda_{1},\dots,\lambda_{m}}\right)\in\mathbb{R}^{m}\,:\,0\leq\lambda_{1},\dots,\lambda_{m}\leq 1,\,\sum_{i=1}^{m}\lambda_{i}w_{i}=1}\right\}. Then, by definition, {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=F\mathopen{}\mathclose{{}\left(P}\right), and hence a polytope as well. ∎

For our next observation we need the following notation. Given $x_{1},\dots,x_{m}\in\mathbb{R}^{n}$, denote the Minkowski sum of the segments \mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left[0,x_{i}}\right]}\right\}_{i=1}^{m} by Z\mathopen{}\mathclose{{}\left(x_{1},\dots,x_{m}}\right). That is,

[TABLE]

In the following proposition, we show that given a measure $\mu=\sum_{i=1}^{m}w_{i}\delta_{x_{i}}$, its generated metronoid is always contained in the intersection of {\rm conv}\mathopen{}\mathclose{{}\left(x_{1},\dots x_{m}}\right) and the zonotope Z\mathopen{}\mathclose{{}\left(w_{1}x_{1},\dots,w_{m}x_{m}}\right).

Proposition 2.4**.**

Let $x_{1},\dots x_{m}\in\mathbb{R}^{n}$, $w_{1},\dots,w_{m}>0$, and set $\mu=\sum_{i=1}^{m}w_{i}\delta_{x_{i}}$. Then

[TABLE]

Proof.

Recall that {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=\mathopen{}\mathclose{{}\left\{\sum_{i=1}^{m}\lambda_{i}w_{i}x_{i}\,:\,0\leq\lambda_{i}\leq 1\>,\>\sum_{i=1}^{m}\lambda_{i}w_{i}=1}\right\}. Then, on the one hand, we may relax the first constraint and obtain that

[TABLE]

On the other hand, we may remove the second constraint and obtain that

[TABLE]

Therefore, we clearly have that {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq P\cap Z. ∎

A picture demonstrating Proposition 2.4 is given in Figure 2.1 below in the particular case where \mu=\delta_{0}+\sum_{i=1}^{2}\frac{1}{k}\mathopen{}\mathclose{{}\left(\delta_{e_{i}}+\delta_{-e_{i}}}\right).

We remark that in Figure 2.1, we have that {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=Z\mathopen{}\mathclose{{}\left(\mu}\right)\cap P\mathopen{}\mathclose{{}\left(\mu}\right) for all values of $k$. However, this is not always the case. For example, consider \mu=\sum_{i=1}^{2}\frac{1}{4}\mathopen{}\mathclose{{}\left(\delta_{e_{i}}+\delta_{e_{-i}}}\right) on $\mathbb{R}^{2}$. Then \mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{2}}\right)=1, and hence {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=\mathopen{}\mathclose{{}\left\{0}\right\}\neq Z\mathopen{}\mathclose{{}\left(\mu}\right)\cap P\mathopen{}\mathclose{{}\left(\mu}\right).

2.1.3. **Zonoid generating measures **

Proposition 2.4 can be stated in a more general case. Given a Borel measure $\mu$ on $\mathbb{R}^{n}$, define

[TABLE]

Then, the same argument verbatim as in the proof of Proposition 2.5 yields:

Proposition 2.5**.**

We have that {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq Z\mathopen{}\mathclose{{}\left(\mu}\right)\cap P\mathopen{}\mathclose{{}\left(\mu}\right).

Remark 2.6*.*

To complement Proposition 2.5, let $\mu$ be a finite Borel measure satisfying that \mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right)\leq 2 and \mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{0}\right\}}\right)\geq 1. We claim that in this case {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=Z\mathopen{}\mathclose{{}\left(\mu}\right). Indeed, note that for any function $0\leq f\leq 1$, \int_{\mathbb{R}^{n}}f\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)\leq\mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{0}\right\}}\right)f\mathopen{}\mathclose{{}\left(0}\right)+1. Hence, by changing the value of f\mathopen{}\mathclose{{}\left(0}\right) (which does not affect \int_{\mathbb{R}^{n}}xf\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)), we may assume that \int_{\mathbb{R}^{n}}f\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=1. Therefore, it follows that, under these assumptions, {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=Z\mathopen{}\mathclose{{}\left(\mu}\right). This fact is also demonstrated in Figure 2.1 above, for \mu=\delta_{0}+\sum_{i=1}^{2}\frac{1}{k}\mathopen{}\mathclose{{}\left(\delta_{e_{i}}+\delta_{-e_{i}}}\right) and $\frac{1}{4}\leq k\leq\frac{1}{2}$.

The next proposition shows that by adding symmetricity to the measures described in Remark 2.6, the generated metronoids become zonoids:

Proposition 2.7**.**

Suppose $\mu$ is a symmetric Borel measure satisfying that $\mu(\mathbb{R}^{n})\leq 2$ , and \mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{0}\right\}}\right)\geq 1. Then

[TABLE]

Proof.

Fix $\theta\in\mathbb{S}^{n-1}.$ Recall the definition of $R_{\theta}$ and $f_{\theta}$ in (2.1), and (2.6). Observe that since $\mu$ is symmetric, \mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right)\leq 2, and \mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{0}\right\}}\right)\geq 1, it follows that $R_{\theta}=0$. Therefore, Proposition 2.1 implies that

[TABLE]

∎

2.1.4. Uniform measures on convex bodies

Let $K\subseteq\mathbb{R}^{n}$ be a convex body, and fix 0<\delta<\text{vol}\mathopen{}\mathclose{{}\left(K}\right). Let $\mu_{\delta}$ be the uniform measure on $K$, defined by $\,{\rm d}\mu_{\delta}=\delta^{-1}\mathbbm{1}_{K}\,{\rm d}x$. Then, for any direction $\theta\in\mathbb{S}^{n-1}$, Proposition 2.1 tells us that h_{{\rm M}\mathopen{}\mathclose{{}\left(\mu_{\delta}}\right)}\mathopen{}\mathclose{{}\left(\theta}\right)=\langle y_{\theta},\,\theta\rangle where

[TABLE]

and R\mathopen{}\mathclose{{}\left(\theta}\right) is the real number satisfying that \text{vol}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{x\in K\,:\,\langle x,\,\theta\rangle\geq R\mathopen{}\mathclose{{}\left(\theta}\right)\}}\right\}}\right)=\delta.

The body {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) is related to the floating body K_{\delta}=\bigcap_{\theta\in\mathbb{S}^{n-1}}\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{n}\,:\,\langle x,\,\theta\rangle\leq R\mathopen{}\mathclose{{}\left(\theta}\right)}\right\} in the following sense: the boundary points of {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) are the centers of mass of the caps \mathopen{}\mathclose{{}\left\{x\in K:\,\langle x,\,\theta\rangle\geq R\mathopen{}\mathclose{{}\left(\theta}\right)}\right\} which are cut off in order to obtain $K_{\delta}$ (see [26] for more about floating bodies). In fact, one can show that K_{\delta}\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu_{\delta}}\right)\subseteq K_{\frac{\delta}{e}}.

2.2. Some linear-invariance properties

In this section we state a few basic facts concerning the behavior of metronoids under linear transformations, and the invariance of the quantities d_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right), D_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right), and {\rm vein^{*}}\mathopen{}\mathclose{{}\left(K}\right).

Let T\in{\rm GL}_{n}\mathopen{}\mathclose{{}\left(\mathbb{R}}\right) be an invertible linear transformation on $\mathbb{R}^{n}$. Given a Borel measure $\mu$ on $\mathbb{R}^{n}$, denote by $\nu=T\#\mu$ the pushforward of $\mu$ by $T$, that is \nu\mathopen{}\mathclose{{}\left(A}\right)=\mu\mathopen{}\mathclose{{}\left(T^{-1}A}\right) for any Borel set $A\subseteq\mathbb{R}^{n}$. Then we have:

Fact 2.8**.**

Let $\mu$ be a Borel measure on $\mathbb{R}^{n}$, T\in{\rm GL}_{n}\mathopen{}\mathclose{{}\left(\mathbb{R}}\right), and denote $\nu=T\#\mu$. Then {\rm M}\mathopen{}\mathclose{{}\left(\nu}\right)=T{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). Moreover, for any convex body $K\subseteq\mathbb{R}^{n}$ containing the origin in its interior, we have that \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{TK}\,{\rm d}\nu\mathopen{}\mathclose{{}\left(x}\right).

Proof.

Let x\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). Then x=\int_{\mathbb{R}^{n}}yf\mathopen{}\mathclose{{}\left(y}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(y}\right) for some $0\leq f\leq 1$ with $\int_{\mathbb{R}^{n}}f\,{\rm d}\mu=1$, and hence

[TABLE]

Similarly, if z\in{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right) then z=\int_{\mathbb{R}^{n}}yg\mathopen{}\mathclose{{}\left(y}\right)\,{\rm d}\nu\mathopen{}\mathclose{{}\left(y}\right) for some $0\leq g\leq 1$ with $\int_{\mathbb{R}^{n}}g\,{\rm d}\nu=1$, and hence

[TABLE]

Let $K\subseteq\mathbb{R}^{n}$ be a convex body containing the origin in its interior. Then

[TABLE]

∎

Fact 2.9**.**

Let $K$ be a convex body in $\mathbb{R}^{n}$, T\in{\rm GL}_{n}\mathopen{}\mathclose{{}\left(\mathbb{R}}\right), and $R\geq 1$. Then d_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right)=d_{R}^{*}\mathopen{}\mathclose{{}\left(TK}\right), D_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right)=D_{R}^{*}\mathopen{}\mathclose{{}\left(TK}\right), and {\rm vein^{*}}\mathopen{}\mathclose{{}\left(K}\right)={\rm vein^{*}}\mathopen{}\mathclose{{}\left(TK}\right).

Proof.

Let $\mu$ be a measure such that K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq RK, and let T\in{\rm GL}_{n}\mathopen{}\mathclose{{}\left(\mathbb{R}}\right). Then by considering the pushforward measure $\nu=T\#\mu$. By Fact 2.8, we have that {\rm M}\mathopen{}\mathclose{{}\left(\nu}\right)=T{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right), and hence TK\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right)\subseteq R\mathopen{}\mathclose{{}\left(TK}\right). Moreover, we clearly have that \nu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right)=\mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right), from which it follows that d_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right)=d_{R}^{*}\mathopen{}\mathclose{{}\left(TK}\right). Finally, note that Fact 2.8 also implies that \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{TK}\,{\rm d}\nu\mathopen{}\mathclose{{}\left(x}\right)=\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right), which means that D_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right)=D_{R}^{*}\mathopen{}\mathclose{{}\left(TK}\right), as required. ∎

2.3. Approximations by discrete measures

In this section we show that, for the purpose of approximating a convex body $K\subseteq\mathbb{R}^{n}$, one can often replace a general Borel measure $\mu$ by a finite discrete measure, without increasing the cost \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right).

We begin with the reduction of infinite measures to finite measures:

Lemma 2.10**.**

Let $K\subseteq\mathbb{R}^{n}$ be a convex body containing [math] in its interior, and $\mu$ be an infinite Borel measure such that K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right), and \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)<\infty. Then for any $\varepsilon>0$, there exists a finite Borel measure $\nu$ such that {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right)\subseteq\mathopen{}\mathclose{{}\left(1+\varepsilon}\right){\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) and, in particular, K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right). Furthermore, we also have that \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\nu\mathopen{}\mathclose{{}\left(x}\right)\leq(1+\varepsilon)\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right).

Proof.

First, we show that we can reduce to the case where \mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{0}\right\}}\right)<\infty. Indeed, suppose \mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{0}\right\}}\right)=\infty, and define a measure $\nu$ by setting \nu\mathopen{}\mathclose{{}\left(A}\right)=\mu\mathopen{}\mathclose{{}\left(A\backslash\mathopen{}\mathclose{{}\left\{0}\right\}}\right) for any measurable set $A$. Let y\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right), and $0\leq f\leq 1$ be a function such that \int_{\mathbb{R}^{n}}f\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=1 and y=\int_{\mathbb{R}^{n}}xf\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right). The conditions \mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{0}\right\}}\right)=\infty and \int_{\mathbb{R}^{n}}f\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=1 force f\mathopen{}\mathclose{{}\left(0}\right)=0. Thus, \int_{\mathbb{R}^{n}}f\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\nu\mathopen{}\mathclose{{}\left(x}\right)=1 and y=\int_{\mathbb{R}^{n}}xf\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\nu\mathopen{}\mathclose{{}\left(x}\right), which implies that y\in{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right). On the other hand, let y^{\prime}\in{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right), and $0\leq f\leq 1$ be a function satisfying that \int_{\mathbb{R}^{n}}f\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\nu\mathopen{}\mathclose{{}\left(x}\right)=1 and y^{\prime}=\int_{\mathbb{R}^{n}}xf\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\nu\mathopen{}\mathclose{{}\left(x}\right). Since \mathopen{}\mathclose{{}\left\{0}\right\} is not in the support of $\nu$, we may assume without loss of generality that f\mathopen{}\mathclose{{}\left(0}\right)=0. Hence, \int_{\mathbb{R}^{n}}f\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=1 and y^{\prime}=\int_{\mathbb{R}^{n}}xf\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right), which implies that y^{\prime}\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). Furthermore, we have that \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\nu\mathopen{}\mathclose{{}\left(x}\right)=\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right). Thus, from now on we may assume that \mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{0}\right\}}\right)<\infty.

By Fact 2.8, for any T\in GL_{n}\mathopen{}\mathclose{{}\left(\mathbb{R}}\right), we have that K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\iff TK\subseteq{\rm M}\mathopen{}\mathclose{{}\left(T\#\mu}\right), and \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|{}_{K}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|{}_{TK}\,{\rm d}\mathopen{}\mathclose{{}\left(T\#\mu}\right)\mathopen{}\mathclose{{}\left(x}\right). Therefore, we may assume without loss of generality that B_{2}^{n}\subseteq K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right).

Define the measure $\nu$ by:

[TABLE]

where $\lambda>0$ is a parameter that will be determined later. Since $K\subseteq CB_{2}^{n}$ for some $C>0$, we have that

[TABLE]

Hence we have that $\nu$ is a finite measure, and \mu$$\mathopen{}\mathclose{{}\left(\lambda B_{2}^{n}}\right)=\infty.

Let y\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). Then there exists a function $0\leq f\leq 1$ such that $\int_{\mathbb{R}^{n}}f\,{\rm d}\mu=1$ and y=\int_{\mathbb{R}^{n}}xf\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right). Let $a=1-\int_{(\lambda B_{2}^{n})^{c}}f(x)\,{\rm d}\mu(x)$, and define the function:

[TABLE]

Then $0\leq g\leq 1$ and $\int_{\mathbb{R}^{n}}g(x)\,{\rm d}\nu(x)=1$. Denoting $y^{\prime}=\int xg(x)\,{\rm d}\nu(x)$, we have that

[TABLE]

Similarly, for any y^{\prime}\in{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right) there exists y\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) such that $\|y-y^{\prime}\|_{B_{2}^{n}}\leq\lambda$. Indeed, let $0\leq g\leq 1$ be a function such that \int_{\mathbb{R}^{n}}g\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\nu\mathopen{}\mathclose{{}\left(x}\right)=1 and y^{\prime}=\int_{\mathbb{R}^{n}}xg\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\nu\mathopen{}\mathclose{{}\left(x}\right). To define a corresponding function f\mathopen{}\mathclose{{}\left(x}\right), fix some $0<s<\lambda$ so that 1\leq\mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{x\,:\,s\leq\mathopen{}\mathclose{{}\left\|x}\right\|_{B_{2}^{n}}<\lambda}\right\}}\right)<\infty. The second inequality holds for any $s>0$. If there is no $s>0$ such that the first inequality is satisfied, then \mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{x\,:\,0<\mathopen{}\mathclose{{}\left\|x}\right\|_{B_{2}^{n}}\leq\lambda}\right\}}\right)\leq 1, which together with \mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{0}\right\}}\right)<\infty, contradicts the fact that \mu$$\mathopen{}\mathclose{{}\left(\lambda B_{2}^{n}}\right)=\infty. Define

[TABLE]

Since $0\leq 1-\int_{(\lambda B_{2}^{n})^{c}}g(x)\,{\rm d}\nu(x)\leq 1$, it follows that \text{0\leq\frac{1-\int_{(\lambda B_{2}^{n})^{c}}f(x),{\rm d}\nu(x)}{\mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left{x,:,s\leq\mathopen{}\mathclose{{}\left|x}\right|{B{2}^{n}}<\lambda}\right}}\right)} }\leq 1, and hence $0\leq f\leq 1$. Moreover,

[TABLE]

Denoting y=\int_{\mathbb{R}^{n}}xf\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)\in\text{{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)}, it follows that

[TABLE]

To show the inclusion \mathopen{}\mathclose{{}\left(1-\lambda}\right){\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right), let

[TABLE]

denote the polar body of {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). By the properties of polarity, for any z\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)^{\circ}, there exists y\in M\mathopen{}\mathclose{{}\left(\mu}\right) such that $\langle z,\,y\rangle=1$. Moreover, by the previous argument, there exists y^{\prime}\in{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right) such that \mathopen{}\mathclose{{}\left\|y-y^{\prime}}\right\|_{B_{2}^{n}}\leq\lambda. Therefore, we have that

[TABLE]

where the last inequality is due to the fact that \mathopen{}\mathclose{{}\left\|z}\right\|_{2}\leq 1, as {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)^{\circ}\subseteq B_{2}^{n}. Thus, it follows that \mathopen{}\mathclose{{}\left(1-\lambda}\right){\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right).

For the opposite inclusion, we use the fact that for every y^{\prime}\in{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right), there exists y\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) such that \mathopen{}\mathclose{{}\left\|y-y^{\prime}}\right\|_{B_{2}^{n}}\leq\lambda. Equivalently, {\rm M}\mathopen{}\mathclose{{}\left(\nu}\right)\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)+\lambda B_{2}^{n}. Since B_{2}^{n}\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right), it follows that {\rm M}\mathopen{}\mathclose{{}\left(\nu}\right)\subseteq\mathopen{}\mathclose{{}\left(1+\lambda}\right){\rm M}\mathopen{}\mathclose{{}\left(\mu}\right), and so

[TABLE]

Moreover, by the definition of $\nu$ we have that

[TABLE]

Finally, consider the pushforward measure $\widetilde{\nu}=\frac{I_{n}}{1-\lambda}\#\nu$. By Fact 2.8, we have that

[TABLE]

and, in particular, K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\tilde{\nu}}\right). Furthermore, we have that

[TABLE]

By choosing a sufficiently small $\lambda$, the proof is complete. ∎

The next lemma shows that any finite measure can be replaced with a discrete one:

Lemma 2.11**.**

Let $K\subseteq\mathbb{R}^{n}$ be a convex body containing [math] in its interior, and $\mu$ be a finite Borel measure such that K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right), and \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)<\infty. Then for any $\varepsilon>0$, there exists a finite discrete measure $\nu$ such that {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right)\subseteq\frac{1+2\varepsilon}{1-2\varepsilon}{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) and, in particular, K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right). Moreover,

[TABLE]

Proof.

As in the proof of the previous lemma, we may assume without loss of generality that $B_{2}^{n}\subseteq K$. Fix $\varepsilon>0$, and fix some large $R\in\mathbb{N}$ so that \int_{\mathopen{}\mathclose{{}\left(RB_{\infty}^{n}}\right)^{c}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)\leq\varepsilon, where $B_{\infty}^{n}$ denotes the $n$-cube \mathopen{}\mathclose{{}\left[-1,1}\right]^{n}\subseteq\mathbb{R}^{n}. For any $m\in\mathbb{N}$, let $A_{m}\subset\mathbb{R}^{n}$ be the collection of points

[TABLE]

For each $a\in A_{m}$, we define the box $B_{a}$ by

[TABLE]

and observe that $\{B_{a}\}_{a\in A_{m}}$ is a partition of E:=\mathopen{}\mathclose{{}\left[-\mathopen{}\mathclose{{}\left(R+\frac{1}{2^{m+1}}}\right),\mathopen{}\mathclose{{}\left(R+\frac{1}{2^{m+1}}}\right)}\right)^{n}. Fix a large enough $m$ so that for each $a\in A_{m}$ and every $x,y\in B_{a}$, we have that \mathopen{}\mathclose{{}\left\|x-y}\right\|{}_{B_{2}^{n}}<\frac{\varepsilon}{\mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right)}<\varepsilon. Define the measure

[TABLE]

We claim that for every y\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right), there exists y^{\prime}\in{\rm M}\mathopen{}\mathclose{{}\left(\mu_{m}}\right) such that \mathopen{}\mathclose{{}\left\|y-y^{\prime}}\right\|_{B_{2}^{n}}\leq 2\varepsilon. Indeed, let y\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right), and $0\leq f\leq 1$ be a function such that \int_{\mathbb{R}^{n}}f\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=1 and y=\int_{\mathbb{R}^{n}}xf\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right). Correspondingly, we define the function $g$ with support in $A_{m}$ by setting

[TABLE]

One can verify that $0\leq g\leq 1$. Moreover, we have that

[TABLE]

Thus, y^{\prime}:=\int_{\mathbb{R}^{n}}xg\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)\in{\rm M}\mathopen{}\mathclose{{}\left(\mu_{m}}\right). A direct computation shows that

[TABLE]

The reverse statement is also true. Namely, for any y^{\prime}\in{\rm M}\mathopen{}\mathclose{{}\left(\mu_{m}}\right) there exists y\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) such that \mathopen{}\mathclose{{}\left\|y-y^{\prime}}\right\|_{B_{2}^{n}}\leq 2\varepsilon. Indeed, let y^{\prime}\in{\rm M}\mathopen{}\mathclose{{}\left(\mu_{m}}\right), and $0\leq g\leq 1$ be a function such that \int_{\mathbb{R}^{n}}g\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu_{m}\mathopen{}\mathclose{{}\left(x}\right)=1 and \int_{\mathbb{R}^{n}}g\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu_{m}\mathopen{}\mathclose{{}\left(x}\right)=y^{\prime}. Correspondingly, we define the function $f$ by setting

[TABLE]

Clearly, 0\leq f\mathopen{}\mathclose{{}\left(x}\right)\leq 1. Moreover,

[TABLE]

Setting y=\int_{\mathbb{R}^{n}}xf\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu_{m}\mathopen{}\mathclose{{}\left(x}\right), we obtain that

[TABLE]

Using the same argument as in the proof of Lemma 2.10, one can verify that

[TABLE]

On the other hand, \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|{}_{K}\,{\rm d}\mu_{m}(x)\leq\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|{}_{K}\,{\rm d}\mu(x)+\varepsilon is straightforward if one breaks down the integration to small partitions $B_{a}$ and $E^{c}$.

By replacing $\mu_{m}$ with the pushforward measure $\nu=\frac{I_{n}}{1-2\varepsilon}\#\mu_{m}$, it follows from Fact 2.8 that

[TABLE]

and, in particular, K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right). Furthermore,

[TABLE]

∎

2.4. Scaling effect on discrete measures

Another property that we shall use in the sequel is the following behavior of metronoids that are generated by discrete measures, under scaling:

Proposition 2.12**.**

Let $x_{1},\dots,x_{m}\in\mathbb{R}^{n}$, $a_{1},\dots,a_{m}\geq 0$, and $\mu=\sum_{i=0}^{m}a_{i}\delta_{x_{i}}$, where $x_{0}=0$, and $a_{0}\geq 0$ . Then for any choice of $r_{1},\dots,r_{m}\geq 1$, the measure $\nu=\sum_{i=1}^{m}\frac{a_{i}}{r_{i}}\delta_{r_{i}x_{i}}+\delta_{0}$ satisfies that {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right), where equality holds whenever $\sum_{i=1}^{m}a_{i}\leq 1$ and $a_{0}\geq 1$. Moreover, for any convex body $K\subseteq\mathbb{R}^{n}$ containing [math], we have that \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|{}_{K}\,{\rm d}\mu(x)=\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|{}_{K}\,{\rm d}\nu(x).

Proof.

Let y\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). Then there exists a function $0\leq f\leq 1$ such that $\int_{\mathbb{R}^{n}}f(x)\,{\rm d}\mu(x)=1$, and $y=\int_{\mathbb{R}^{n}}xf(x)\,{\rm d}\mu(x)$. We construct a function $g$, with support on \mathopen{}\mathclose{{}\left\{0,r_{1}x_{1},\cdots,r_{m}x_{m}}\right\}, as follows; $g(r_{i}x_{i}):=f(x_{i})$ for i\in\mathopen{}\mathclose{{}\left\{1,\dots,m}\right\}, and $g(0)=1-\sum_{i=1}^{m}\frac{a_{i}}{r_{i}}g(r_{i}x_{i})$. One can easily verify that 0\leq g\mathopen{}\mathclose{{}\left(r_{i}x_{i}}\right)\leq 1 for all i\in\mathopen{}\mathclose{{}\left\{1,\dots,m}\right\}, and that also 0\leq g\mathopen{}\mathclose{{}\left(0}\right)\leq 1, due to the fact that $r_{i}\geq 1$. Moreover, we have that

[TABLE]

and

[TABLE]

as claimed.

Next, assume that $\sum_{i=1}^{m}a_{i}\leq 1$ and $a_{0}\geq 1$. Let z\in{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right). Then there exists a function $0\leq f\leq 1$ such that $\sum_{i=0}^{m}\frac{a_{i}}{r_{i}}f(r_{i}x_{i})=1$ and $\sum_{i=0}^{m}a_{i}f(r_{i}x_{i})=z$. We define a new function $0\leq g\leq 1$ whose support is \mathopen{}\mathclose{{}\left\{{0,x_{1},\cdots,x_{n}}}\right\} by setting $g(x_{i}):=f(r_{i}x_{i})$ for i\in\mathopen{}\mathclose{{}\left\{1,\dots,m}\right\}, and g\mathopen{}\mathclose{{}\left(0}\right)=1-\sum_{i=1}^{m}a_{i}g(x_{i}). Thus, we have that $0\leq g\leq 1$, $\sum_{i=0}^{m}a_{i}g(x_{i})=1$, and $\sum_{i=0}^{m}a_{i}g(x_{i})x_{i}=z$. Therefore, {\rm M}\mathopen{}\mathclose{{}\left(\nu}\right)\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right), and hence {\rm M}\mathopen{}\mathclose{{}\left(\nu}\right)={\rm M}\mathopen{}\mathclose{{}\left(\mu}\right).

Finally, let $K\subseteq\mathbb{R}^{n}$ be a convex body containing [math]. Since for any $r\geq 0$ and $x\in\mathbb{R}^{n}$, we have that \mathopen{}\mathclose{{}\left\|rx}\right\|{}_{K}=r\mathopen{}\mathclose{{}\left\|x}\right\|{}_{K}, it follows that

[TABLE]

∎

The following observation is an immediate consequence of Proposition 2.12:

Corollary 2.13**.**

Let $K\subseteq\mathbb{R}^{n}$ be a convex body containing [math]. Suppose $\mu$ is a finite discrete measure such that K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). Then there exists a discrete probability measure $\nu$ such that \text{K\subseteq}{\rm M}\mathopen{}\mathclose{{}\left(\nu+\delta_{0}}\right), and \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|{}_{K}\,{\rm d}\mu(x)=\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|{}_{K}\,{\rm d}\nu(x).

Proof.

Suppose $\mu=\sum_{i=0}^{m}a_{i}\delta_{x_{i}}$ with $a_{i}\geq 0$ and $x_{0}=0$, and let r=\mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right)=\sum_{i=0}^{m}a_{i}\geq 1. Then Proposition 2.12 implies that the measure $\nu=\sum\frac{a_{i}}{r}\delta_{rx_{i}}$ satisfies our claim. ∎

3. Estimating d_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right) and D_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right)

In this section we prove Theorems 1.2 and 1.3.

3.1. Proof of Theorem 1.2

Let $\sigma_{r}$ denote the uniform probability measure on $r\mathbb{S}^{n-1}$. For $r=1$ we simply denote $\sigma=\sigma_{1}$.

Proof of Theorem 1.2.

Let $K\subseteq\mathbb{R}^{n}$ be a centrally-symmetric convex body such that $B_{2}^{n}$ is the minimal volume circumscribed ellipsoid of $K$. By John’s theorem (see e.g., [1]), we have that $\frac{1}{\sqrt{n}}B_{2}^{n}\subseteq K\subseteq B_{2}^{n}$.

Consider the measure $\mu=2\sigma_{R}$, where R=\mathopen{}\mathclose{{}\left(\int_{\mathbb{S}^{n-1}}\mathopen{}\mathclose{{}\left|\langle x,\,\theta\rangle}\right|\,{\rm d}\sigma\mathopen{}\mathclose{{}\left(x}\right)}\right)^{-1}. Let $e_{1}\in S^{n-1}$, and define f\mathopen{}\mathclose{{}\left(x}\right)=\mathbbm{1}_{\mathopen{}\mathclose{{}\left\{y\,:\,\langle y,\,e_{1}\rangle>0}\right\}}\mathopen{}\mathclose{{}\left(x}\right). Then we have \int_{\mathbb{R}^{n}}f\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=1, and therefore,

[TABLE]

which implies that B_{2}^{n}\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). In fact, Proposition 2.1 tells us that h_{M\mathopen{}\mathclose{{}\left(\mu}\right)}\mathopen{}\mathclose{{}\left(e_{1}}\right)=1, which means that {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=B_{2}^{n}, and hence \frac{1}{\sqrt{n}}{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right).

Finally, note that \mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right)=2. Moreover, by a standard computation, one can verify that

[TABLE]

Therefore,

[TABLE]

which completes our proof. ∎

3.2. Proof of Theorem 1.3

To prove Theorem 1.3, we need the following consequence of the Brunn-Minkowski theorem which was observed (in equivalent forms) several times in the literature, e.g., in [20], and [27]. For the sake of completeness, we provide a proof.

Proposition 3.1**.**

Let $K\subseteq\mathbb{R}^{n}$ be a centered convex body. Fix $R>1$, $u\in\mathbb{S}^{n-1}$, and r=h_{K}\mathopen{}\mathclose{{}\left(u}\right). Let L:=K\cap\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{n}\,:\,\langle x,\,u\rangle\geq\frac{r}{R}}\right\}. Then

[TABLE]

To prove Proposition 3.1, we need the following lemma. Given $R>1$, and a non-negative concave function f:\mathopen{}\mathclose{{}\left[0,R}\right]\rightarrow\mathbb{R}, let \widetilde{f}:\mathopen{}\mathclose{{}\left[0,R}\right]\to\mathbb{R} denote the linear function satisfying \tilde{f}\mathopen{}\mathclose{{}\left(1}\right)=f\mathopen{}\mathclose{{}\left(1}\right) and \tilde{f}\mathopen{}\mathclose{{}\left(R}\right)=0.

Lemma 3.2**.**

For $R>1$, let f:\mathopen{}\mathclose{{}\left[0,R}\right]\rightarrow\mathbb{\mathbb{R}} be a non-negative concave function. Then for any increasing function g:\mathopen{}\mathclose{{}\left[0,\infty}\right)\rightarrow\mathopen{}\mathclose{{}\left[0,\infty}\right), we have

[TABLE]

Proof of Lemma 3.2 .

Let $A_{1}:=\int_{0}^{1}g\circ f(t)\,{\rm d}t$ and $A_{2}:=\int_{1}^{R}g\circ f(t)\,{\rm d}t$. In particular,

[TABLE]

Our next goal is to bound $A_{1}$ from above and $A_{2}$ from below. To bound $A_{1}$ from above, note that since $f$ is concave and non-negative, we have that for any $t\in[0,1]$,

[TABLE]

Since $g$ is increasing, we thus obtain

[TABLE]

Similarly, we bound $A_{2}$ from above by noting that for any $t\in[1,R]$,

[TABLE]

and hence

[TABLE]

Finally, the above bounds for $A_{1}$ and $A_{2}$ imply that

[TABLE]

∎

Proof of Proposition 3.1 .

We may rescale $K$ so that $r=R$. Let

[TABLE]

and $f(t):=\text{vol}(K_{t})^{\frac{1}{n-1}}$. By Brunn-Minkowski theorem, $f(t)$ is a concave function on its support. Moreover, we clearly have that $\text{vol}(K_{+})=\int_{0}^{R}f^{n-1}(t)\,{\rm d}t$ and $\text{vol}(L)=\int_{1}^{R}f^{n-1}(t)\,{\rm d}t$. Also note that $\tilde{f}(t)=\frac{f(1)}{1-\frac{1}{R}}(1-\frac{t}{R})$. Therefore, Lemma 3.2, applied with f\mathopen{}\mathclose{{}\left(t}\right) and g\mathopen{}\mathclose{{}\left(t}\right)=t^{n-1}, implies that

[TABLE]

where the last inequality relies on the fact that $1+x\leq e^{x}$. Since, by Grünbaum [13], we know that $\frac{\text{vol}(K_{+})}{\text{vol}(K)}\geq\frac{1}{e}$, our proof is complete. ∎

We are now ready to prove Theorem 1.3:

Proof of Theorem 1.3 .

Let $K\subseteq\mathbb{R}^{n}$ be a centered convex body, and let $\mu$ be the uniform measure on $RK$, satisfying \,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=\frac{\exp\mathopen{}\mathclose{{}\left(1+\frac{n-1}{R-1}}\right)}{\text{vol}\mathopen{}\mathclose{{}\left(RK}\right)}\mathbbm{1}_{RK}(x)\,{\rm d}x. Define

[TABLE]

By Proposition 3.1, we have that

[TABLE]

In particular, we get that \mu\mathopen{}\mathclose{{}\left(L_{\theta}}\right)=\frac{\exp\mathopen{}\mathclose{{}\left(1+\frac{n-1}{R-1}}\right)}{\text{vol}\mathopen{}\mathclose{{}\left(RK}\right)}\text{vol}\mathopen{}\mathclose{{}\left(L_{\theta}}\right)\geq 1.

Fix $\theta\in\mathbb{S}^{n-1}$, and let f_{\theta}\mathopen{}\mathclose{{}\left(x}\right)=\frac{1}{\mu(L_{\theta})}\mathbf{1}_{L_{\theta}}. By the previous argument, $0\leq f_{\theta}\leq 1$, and $\int_{\mathbb{R}^{n}}f_{\theta}(x)\,{\rm d}\mu(x)=1$. Therefore, x_{\theta}:=\int_{\mathbb{R}^{n}}xf_{\theta}\mathopen{}\mathclose{{}\left(x}\right)\,{\rm d}\mu(x)\in{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). In particular, by the definition of $L_{\theta}$, it follows that

[TABLE]

and therefore K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right).

Finally, we have that d_{R}^{*}\mathopen{}\mathclose{{}\left(K}\right)\leq\mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right)=\exp\mathopen{}\mathclose{{}\left(1+\frac{n-1}{R-1}}\right), and

[TABLE]

Moreover, it follows that {\rm vein^{*}}\mathopen{}\mathclose{{}\left(K}\right)\leq D_{n}^{*}\mathopen{}\mathclose{{}\left(K}\right)\leq e^{2}n. ∎

4. the fractional vertex index

4.1. A couple of extremal examples

4.1.1. The vertex index of the cross-polytope

Proposition 4.1**.**

We have that {\rm vein^{*}}\mathopen{}\mathclose{{}\left(B_{1}^{n}}\right)=2n.

Proof.

We follow the lines of the proof in [5] that {\rm vein}\mathopen{}\mathclose{{}\left(B_{1}^{n}}\right)=2n. Let \mathopen{}\mathclose{{}\left\|\cdot}\right\|_{1} denote the norm induced by $B_{1}^{n}$, that is \mathopen{}\mathclose{{}\left\|x}\right\|_{1}=\sum_{i=1}^{n}\mathopen{}\mathclose{{}\left|x_{i}}\right|=\sum_{i=1}^{n}\mathopen{}\mathclose{{}\left|\langle x,\,e_{i}\rangle}\right|, where $e_{1},\dots,e_{n}$ are the standard basis of $\mathbb{R}^{n}$. Let $\mu$ be a measure such that B_{1}^{n}\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). Then for each $e_{i}$ there exists a function $0\leq f\leq 1$ such that $e_{i}=\int_{\mathbb{R}^{n}}xf(x)\,{\rm d}\mu(x)$, and hence $1=\int_{\mathbb{R}^{n}}\langle x,\,e_{i}\rangle f(x)\,{\rm d}\mu(x)$. Furthermore, if we define the function $g$ by

[TABLE]

we obtain the following inequality: 1\leq\int_{\mathbb{R}^{n}}\langle x,\,e_{i}\rangle g(x)\,{\rm d}\mu(x)\leq\int_{\mathbb{R}^{n}}\max\mathopen{}\mathclose{{}\left\{\langle x,\,e_{i}\rangle,0}\right\}\,{\rm d}\mu(x). Applying the same argument to $-e_{i}$, we have 1\leq\int_{\mathbb{R}^{n}}\max\mathopen{}\mathclose{{}\left\{\langle x,\,-e_{i}\rangle,0}\right\}\,{\rm d}\mu(x). Therefore, it follows that

[TABLE]

On the other hand, if $\mu=\sum_{i=1}^{n}(\delta_{e_{i}}+\delta_{-e_{i}})$, then B_{1}^{n}={\rm M}\mathopen{}\mathclose{{}\left(\mu}\right), and \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|{}_{1}\,{\rm d}\mu(x)=2n. Therefore, the lower bound is attained by $\mu$. ∎

4.1.2. The vertex index of the Euclidean ball

Proposition 4.2**.**

We have that {\rm vein^{*}}\mathopen{}\mathclose{{}\left(B_{2}^{n}}\right)=\sqrt{2\pi n}\mathopen{}\mathclose{{}\left(1+{\rm o}\mathopen{}\mathclose{{}\left(1}\right)}\right).

Proof.

The upper bound {\rm vein^{*}}\mathopen{}\mathclose{{}\left(B_{2}^{n}}\right)\leq\sqrt{2\pi n}\mathopen{}\mathclose{{}\left(1+{\rm o}\mathopen{}\mathclose{{}\left(1}\right)}\right) follows verbatim from the proof of theorem 1.2 by considering the measure $\mu=2\sigma_{R}$, where R=\mathopen{}\mathclose{{}\left(\int_{\mathbb{S}^{n-1}}\mathopen{}\mathclose{{}\left|\langle x,\,\theta\rangle}\right|\,{\rm d}\sigma\mathopen{}\mathclose{{}\left(x}\right)}\right)^{-1}, which implies that {\rm M}\mathopen{}\mathclose{{}\left(\mu}\right)=B_{2}^{n} , and

[TABLE]

Next, we show that {\rm vein^{*}}\mathopen{}\mathclose{{}\left(B_{2}^{n}}\right)\geq\sqrt{2\pi n}\mathopen{}\mathclose{{}\left(1+{\rm o}\mathopen{}\mathclose{{}\left(1}\right)}\right). Let $\mu$ be any measure satisfying that K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). By Lemmas 2.10, 2.11, and Proposition 2.12, we may assume without loss of generality that $\mu$ is discrete, finite, and that {\rm supp}\mathopen{}\mathclose{{}\left(\mu}\right)\subseteq r\mathbb{S}^{n-1}\cup\mathopen{}\mathclose{{}\left\{0}\right\} for some $r>0$. By adding $\delta_{0}$ to $\mu$ at no additional cost, we may also assume that $\mu$ has an atom $D_{0}\delta_{0}$ at the origin.

Let {\rm SO}\mathopen{}\mathclose{{}\left(n}\right) be the rotation group on $\mathbb{R}^{n}$, and let $\xi$ be the normalized probability Haar measure on {\rm SO}\mathopen{}\mathclose{{}\left(n}\right). We define the radial measure $\mu_{0}$ by letting

[TABLE]

for any Borel set $A\subseteq\mathbb{R}^{n}$. Note that $\mu_{0}=D_{0}\delta_{0}+D_{r}\sigma_{r}$, where D_{r}r=\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{B_{2}^{n}}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right). By Proposition 2.2, for any $\theta\in\mathbb{S}^{n-1}$ we have

[TABLE]

Combined with (3.1), the above inequality implies that

[TABLE]

Therefore, it follows that \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{B_{2}^{n}}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)=D_{r}r\geq\sqrt{2\pi n}\mathopen{}\mathclose{{}\left(1+{\rm o}\mathopen{}\mathclose{{}\left(1}\right)}\right), as claimed. ∎

Remark 4.3*.*

A shorter argument provides the slightly worst lower bound {\rm vein^{*}}\mathopen{}\mathclose{{}\left(B_{2}^{n}}\right)\geq 2\sqrt{n}. Indeed, since d\mathopen{}\mathclose{{}\left(B_{2}^{n},\,B_{1}^{n}}\right)=\sqrt{n}, Proposition 4.1 and Fact 4.4 imply that

[TABLE]

4.2. A Lower bound

This section is devoted for the proof of Theorem 1.5.

We will need the following fact which relates the fractional vertex index of two convex bodies through their Banach-Mazur distance. Let $I_{n}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ denote the identity operator on $\mathbb{R}^{n}$. Let d\mathopen{}\mathclose{{}\left(K,L}\right) denote the Banach-Mazur distance between two centrally-symmetric convex bodies, $K,L\subseteq\mathbb{R}^{n}$. In [5], the authors show that {\rm vein}\mathopen{}\mathclose{{}\left(K}\right)\leq{\rm vein}\mathopen{}\mathclose{{}\left(L}\right)d\mathopen{}\mathclose{{}\left(K,L}\right). Analogously, we have:

Fact 4.4**.**

Let $K,L$ be centrally-symmetric convex bodies in $\mathbb{R}^{n}$. Then

[TABLE]

Proof.

Let $T$ be some invertible linear transformation such that TL\subseteq K\subseteq d\mathopen{}\mathclose{{}\left(K,L}\right)TL. Suppose $\mu$ is a measure satisfying that TL\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). Then, by Fact 2.8, K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(d(K,L)\cdot I_{n}\#\mu}\right) and hence

[TABLE]

Since {\rm vein^{*}}\mathopen{}\mathclose{{}\left(L}\right) is linear-invariant, and $\mu$ is arbitrary, it follows that

[TABLE]

∎

We shall also use the following proportional Dvoretzky-Rogers factorization Theorem by Bourgain and Szarek:

Theorem** (Bourgain and Szarek [14]).**

If \mathopen{}\mathclose{{}\left(X,\|\cdot\|}\right) is an $n$-dimensional normed space and $\epsilon\in(0,1)$, there exists vectors $x_{1},\dots,x_{m}\in X$, $m\geq(1-\epsilon)n$, such that for any real $t_{1},\dots,t_{n}$,

[TABLE]

where $c,d>0$ are absolute constants.

Let us fix an orthogonal basis \mathopen{}\mathclose{{}\left\{e_{1},\dots,e_{n}}\right\} of $\mathbb{R}^{n}$, and $\epsilon=\frac{1}{2}$. Given a subspace $E\subseteq\mathbb{R}^{n}$ and $1\leq p\leq\infty$, denote $B_{p}^{E}=B_{p}^{n}\cap E$. For our purpose, it will be enough to use the following simpler geometric version of the above proportional Dvoretzky-Rogers factorization Theorem:

Theorem 4.5**.**

Let $K\subset\mathbb{R}^{n}$ be a centrally-symmetric convex body. Let $E\subseteq\mathbb{R}^{n}$ be the subspace spanned by e_{1},\cdots,e_{\mathopen{}\mathclose{{}\left\lceil n/2}\right\rceil}. Then there exists a linear transformation T\in{\rm GL}_{n}\mathopen{}\mathclose{{}\left(\mathbb{R}}\right) such that

[TABLE]

where $c>0$ is a universal constant.

We are now ready to prove Theorem 1.5:

Proof of Theorem 1.5 .

By Theorem 4.5, applied to $K^{\circ}$, and the fact that $B_{\infty}^{n}\subseteq\sqrt{n}B_{2}^{n}$, there exists T\in{\rm GL}{}_{n}\mathopen{}\mathclose{{}\left(\mathbb{R}}\right) such that

[TABLE]

Let ${\rm Pr}_{E}:\mathbb{R}^{n}\to E$ denote the orthogonal projection onto $E$, and set \widetilde{K}=\mathopen{}\mathclose{{}\left(T^{-1}}\right)^{\intercal}K. By the properties of polarity, (4.1) is equivalent to

[TABLE]

In other words, we have that d\mathopen{}\mathclose{{}\left({\rm Pr}_{E}K,\,B_{1}^{E}}\right)\leq\frac{\sqrt{n}}{c} .

Next, fix $\varepsilon>0$. By Lemmas 2.10 and 2.11, there exists a finite discrete measure $\mu$ of the form $\mu=\sum_{i=1}^{N}w_{i}\delta_{x_{i}}$ such that K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right) and \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|_{K}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right)\leq{\rm vein^{*}}\mathopen{}\mathclose{{}\left(K}\right)+\varepsilon. Consider the measure $\nu$ on $E$ defined by \nu=\sum_{i=1}^{N}w_{i}\delta_{{\rm Pr}_{E}\mathopen{}\mathclose{{}\left(x_{i}}\right)}. One can verify that {\rm Pr}_{E}\mathopen{}\mathclose{{}\left(K}\right)\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right). Moreover, since \mathopen{}\mathclose{{}\left\|x}\right\|_{K}\geq\mathopen{}\mathclose{{}\left\|{\rm Pr}_{E}\mathopen{}\mathclose{{}\left(x}\right)}\right\|_{{\rm Pr}_{E}\mathopen{}\mathclose{{}\left(K}\right)} for any $x\in\mathbb{R}^{n}$, it follows that

[TABLE]

On the other hand, by Fact 4.4, we have that

[TABLE]

where we used {\rm vein^{*}}\mathopen{}\mathclose{{}\left(B_{1}^{E}}\right)=2\dim\mathopen{}\mathclose{{}\left(E}\right)=n in the last inequality. ∎

Remark 4.6*.*

Note that the upper bound {\rm vein^{*}}\mathopen{}\mathclose{{}\left(K}\right)\leq e^{2}n, for any convex body $K$, immediately follows from Corollary 1.4.

5. An application to centroid bodies

In this section we show how Corollary 1.7 is a direct consequence of Theorem 1.5.

5.1. Reformulating {\rm vein^{*}}\mathopen{}\mathclose{{}\left(K}\right)

Let $\mu$ be a Borel measure such that K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right). By Lemmas 2.10, 2.11, and Proposition 2.12, we may assume without loss of generality that $\mu$ is a finite discrete measure with \mu\mathopen{}\mathclose{{}\left(\mathbb{R}^{n}}\right)=2 and \mu\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\{0}\right\}}\right)=1. Let $\mathscr{D}_{n}$ be the collection of all non-degenerate finite discrete probability measure. Then, the previous argument implies that

[TABLE]

Moreover, if $K=-K$ then for each measure $\mu$ such that K\subseteq{\rm M}\mathopen{}\mathclose{{}\left(\mu}\right), we can define the symmetric measure $\nu$ by $\nu(A)=\frac{\mu(A)+\mu(-A)}{2}$ for any Borel set $A\subseteq\mathbb{R}^{n}$. Then $\nu$ is a symmetric measure, satisfying that K\subset{\rm M}\mathopen{}\mathclose{{}\left(\nu}\right) and \int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|{}_{K}\,{\rm d}\nu(x)=\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|{}_{K}\,{\rm d}\mu(x). Therefore, we conclude that

[TABLE]

where $\mathscr{D}_{n}^{s}$ is the collection of all symmetric non-degenerate finite discrete probability measures.

In view of Remark 2.6, the above equality immediately implies the following reformulation of the fractional vertex index:

Proposition 5.1**.**

For any convex body $K\subseteq\mathbb{R}^{n}$, we have

[TABLE]

5.2. A relation to $L_{1}$-centroid bodies

Let $\mathcal{K}_{n}$ be the class of all symmetric convex bodies in $\mathbb{R}^{n}$, and $\mathscr{F}_{n}$ the class of all non-degenerate Borel probability measures on $\mathbb{R}^{n}$ with bounded first moment. We have the following equivalence:

Proposition 5.2**.**

{\displaystyle\inf_{K\in\mathcal{K}_{n}}{\rm vein^{*}}\mathopen{}\mathclose{{}\left(K}\right)=2\inf_{\mu\in\mathscr{F}_{n}}\int_{\mathbb{R}^{n}}\mathopen{}\mathclose{{}\left\|x}\right\|{}_{Z_{1}(\mu)}\,{\rm d}\mu\mathopen{}\mathclose{{}\left(x}\right).}**

Proof.

By (5.1), we have that

[TABLE]

Moreover, Proposition 2.7 implies that for any $\mu\in\mathscr{D}_{n}^{s}$, {\rm M}\mathopen{}\mathclose{{}\left(\mu+\delta_{0}}\right)=\frac{1}{2}Z_{1}(\mu). Therefore,

[TABLE]

By observing that

[TABLE]

which does not depend on $K$, we obtain that

[TABLE]

To conclude, we have that

[TABLE]

as claimed. ∎

Finally, note that Corollary 1.7 follows directly from Theorem 1.5 and Proposition 5.2.