Norms of weighted sums of log-concave random vectors

TL;DR

This paper establishes upper bounds for weighted sums of log-concave random vectors, providing an alternative proof for known lower bounds and exploring applications in randomized vector balancing.

Contribution

It introduces new upper bounds for multi-integral norms of weighted sums of log-concave vectors and offers an alternative proof for existing lower bounds, with applications to vector balancing.

Findings

Derived upper bounds for the multi-integral expression in isotropic convex bodies.

Provided an alternative proof for the sharp lower bound by Gluskin and Milman.

Applied results to randomized vector balancing problems.

Abstract

Let $C$ and $K$ be centrally symmetric convex bodies of volume $1$ in ${\mathbb R}^n$. We provide upper bounds for the multi-integral expression \begin{equation*}\|{\bf t}\|_{C^s,K}=\int_{C}\cdots\int_{C}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_1\cdots dx_s\end{equation*} in the case where $C$ is isotropic. Our approach provides an alternative proof of the sharp lower bound, due to Gluskin and V. Milman, for this quantity. We also present some applications to "randomized" vector balancing problems.

Equations

$$\|{\bf t}\|_{C^{s},K}=\int_{C}\cdots\int_{C}\Big{\|}\sum_{j=1}^{s}t_{j}x_{j}\Big{\|}_{K}\,dx_{1}\cdots dx_{s}$$

$$\|{\bf t}\|_{{\cal C},K}=\frac{1}{\prod_{j=1}^{s}{\rm vol}_{n}(C_{j})}\int_{C_{1}}\cdots\int_{C_{s}}\Big{\|}\sum_{j=1}^{s}t_{j}x_{j}\Big{\|}_{K}\,dx_{1}\cdots dx_{s},$$

$$\|{\bf t}\|_{{\cal C},K}\geqslant c\sqrt{s}\Big{(}\prod_{j=1}^{s}|t_{j}|\Big{)}^{1/s}\Big{(}\prod_{j=1}^{s}{\rm vol}_{n}(C_{j})\Big{)}^{\frac{1}{sn}}/{\rm vol}_{n}(K)^{1/n},$$

$$\|{\bf t}\|_{{\cal A},K}:=\frac{1}{\prod_{j=1}^{s}{\rm vol}_{n}(A_{j})}\int_{A_{1}}\cdots\int_{A_{s}}\Big{\|}\sum_{j=1}^{s}t_{j}x_{j}\Big{\|}_{K}\,dx_{1}\cdots dx_{s}\geqslant c\Big{(}\sum_{j=1}^{s}t_{j}^{2}\Big{(}\frac{{\rm vol}_{n}(A_{j})}{{\rm vol}_{n}(K)}\Big{)}^{2/n}\Big{)}^{1/2},$$

$$\Vert \mathbf{t}{\Vert }_{\mathcal{A},K}⩾c\Vert \mathbf{t}{\Vert }_2$$

$$\displaystyle{\rm vol}_{ns}\Big{(}\Big{\{}(x_{j})_{1\leqslant j\leqslant s}:x_{j}\in A_{j}\;\;\hbox{for all}\;j\;\;\hbox{and}\;\Big{\|}\sum_{j=1}^{s}t_{j}x_{j}\Big{\|}_{K}<\alpha\Big{\}}\Big{)}$$

$$\displaystyle\hskip 56.9055pt\leqslant{\rm vol}_{ns}\Big{(}\Big{\{}(x_{j})_{1\leqslant j\leqslant s}:x_{j}\in B_{2}^{n}\;\;\hbox{for all}\;j\;\;\hbox{and}\;\Big{\|}\sum_{j=1}^{s}t_{j}x_{j}\Big{\|}_{2}<\alpha\Big{\}}\Big{)}$$

$$\Vert \mathbf{t}{\Vert }_{\mathcal{C},K}=\Vert \mathbf{t}{\Vert }_2{\int }_{{\mathbb{R}}^n}\Vert x{\Vert }_{K}d{\nu }_{\mathbf{t}}(x),$$

$$\Vert \mathbf{t}{\Vert }_{\mathcal{C},K}⩾\frac{n}{e(n+1)}\Vert \mathbf{t}{\Vert }_2.$$

$$\Vert \mathbf{t}{\Vert }_{C^s,K}=\Vert \mathbf{t}{\Vert }_2{L}_C{I}_1({\mu }_{\mathbf{t}},K),$$

$$I_1(\mu ,K)={\int }_{{\mathbb{R}}^n}\Vert x{\Vert }_{K}d\mu (x).$$

$${\int }_{O(n)}{I}_1(\mu ,U(K))d\nu (U)={\int }_{{\mathbb{R}}^n}{\int }_{O(n)}\Vert x{\Vert }_{U(K)}d\nu (U)d\mu (x)=M(K){\int }_{{\mathbb{R}}^n}\Vert x{\Vert }_{2}d\mu (x)\approx \sqrt{n}M(K),$$

$$M(K):={\int }_{S^{n-1}}\Vert \xi {\Vert }_{K}d\sigma (\xi )$$

$${\int }_{O(n)}\Vert \mathbf{t}{\Vert }_{U(C{)}^s,K}\approx (L_C\sqrt{n}M(K))\Vert \mathbf{t}{\Vert }_2.$$

$$M(K)⩽\frac{c{\log }^{2/5}(e+n)}{\sqrt[10]{n}{L}_K}.$$

$$M(K)⩽\frac{c\sqrt[3]{\varrho }{\log }^{1/3}(e+n)}{\sqrt[6]{n}{L}_K}.$$

$$c_1{L}_{C}R(K^{\circ })\Vert \mathbf{t}{\Vert }_2⩽\Vert \mathbf{t}{\Vert }_{C^s,K}⩽\sqrt{n}{L}_{C}R(K^{\circ })\Vert \mathbf{t}{\Vert }_2,$$

$$\Vert \mathbf{t}{\Vert }_{C^s,K}⩽(c_2{L}_C/\sqrt{\alpha })\Vert \mathbf{t}{\Vert }_2$$

$$\Vert \mathbf{t}{\Vert }_{K^s,K}⩽(c_3/\alpha )\Vert \mathbf{t}{\Vert }_2$$

$$\|{\bf t}\|_{C^{s},K}\leqslant c\,\Big{(}L_{C}\max\Big{\{}\sqrt[4]{n},\sqrt{\log(1+s)}\Big{\}}\Big{)}\,\sqrt{n}M(K)\|{\bf t}\|_{2}$$

$$\Vert \mathbf{t}{\Vert }_{C^s,K}⩽c{\varrho }^2\sqrt{n}M(K)\Vert \mathbf{t}{\Vert }_2$$

$$\frac{c_{3}}{C_{2}(X_{K})}\,\|{\bf t}\|_{2}\leqslant\|{\bf t}\|_{K^{s},K}\leqslant\big{(}c_{4}L_{K}C_{2}(X_{K})\sqrt{n}M(K_{{\rm iso}})\big{)}\,\|{\bf t}\|_{2},$$

$$\Vert \mathbf{t}{\Vert }_{\mathcal{C},K}⩽c\sqrt{\log n}\cdot \max \{\Vert \mathbf{t}{\Vert }_2,\sqrt{\log n}\Vert \mathbf{t}{\Vert }_{\infty }\}$$

$$\Vert \mathbf{t}{\Vert }_{{\overline{B_p^n}}^s,\overline{B_p^n}}⩽c\min \{\sqrt{p},\sqrt{\log n}\}\Vert \mathbf{t}{\Vert }_2$$

$$\beta_{s}(C,K):=\min\Big{\{}r>0:\;\;\hbox{for any}\;x_{1},\ldots,x_{s}\in C,\;\min_{\epsilon\in E_{2}^{s}}\Big{\|}\sum_{j=1}^{s}\epsilon_{j}x_{j}\Big{\|}_{K}\leqslant r\Big{\}},$$

$$\tilde{\beta }(C,K)⩽2\underset{k⩽n}{\max }{\beta }_k(C,K).$$

$$\beta_{\delta,s}^{(R)}(C,K):=\min\Big{\{}r>0:{\rm vol}_{ns}\Big{(}\Big{\{}(x_{j})_{j=1}^{s}:x_{j}\in C\;\hbox{for all}\;j\;\hbox{and}\;\min_{\epsilon\in E_{2}^{s}}\Big{\|}\sum_{j=1}^{s}\epsilon_{j}x_{j}\Big{\|}_{K}\leqslant r\Big{\}}\Big{)}\geqslant 1-\delta\Big{\}}.$$

$${\beta }_{\delta ,n}^{(R)}(K,K)⩽(c\log (2/\delta )L_K{n}^{3/4})\sqrt{n}M(K_{\mathrm{iso}})$$

$${\beta }_{\delta ,n}^{(R)}(K,K)⩽(c\log (2/\delta ){\varrho }^2\sqrt{n})\sqrt{n}M(K_{\mathrm{iso}}).$$

$${\beta }_{\delta ,n}^{(R)}(K,K)⩽(c\log (2/\delta )\sqrt{n}/\alpha ),$$

$${\beta }_{\delta ,n}^{(R)}(K,K)⩽(c\log (2/\delta )L_K{C}_2(X_K)\sqrt{n})\sqrt{n}M(K_{\mathrm{iso}}).$$

Full text

[TABLE]

in the case where $C$ is isotropic. Our approach provides an alternative proof of the sharp lower bound, due to Gluskin and V. Milman, for this quantity. We also present some applications to “randomized” vector balancing problems.

1 Introduction

Let $K$ be a centrally symmetric convex body in ${\mathbb{R}}^{n}$. For any $s$-tuple ${\cal C}=(C_{1},\ldots,C_{s})$ of centrally symmetric convex bodies $C_{j}$ in ${\mathbb{R}}^{n}$ we consider the norm on ${\mathbb{R}}^{s}$, defined by

[TABLE]

where ${\bf t}=(t_{1},\ldots,t_{s})$. If ${\cal C}=(C,\ldots,C)$ then we write $\|{\bf t}\|_{C^{s},K}$ instead of $\|{\bf t}\|_{{\cal C},K}$. A question posed by V. Milman is to determine if, in the case $C=K$, one has that $\|\cdot\|_{K^{s},K}$ is equivalent to the standard Euclidean norm up to a term which is logarithmic in the dimension, and in particular, if under some cotype condition on the norm induced by $K$ to ${\mathbb{R}}^{n}$ one has equivalence between $\|\cdot\|_{K^{s},K}$ and the Euclidean norm.

This question was studied by Bourgain, Meyer, V. Milman and Pajor in [9]; they obtained the lower bound

[TABLE]

where $c>0$ is an absolute constant. Gluskin and V. Milman studied the same question in [17] and obtained a better lower bound in a more general context.

Theorem 1.1** (Gluskin-Milman).**

Let $A_{1},\ldots,A_{s}$ be measurable sets in ${\mathbb{R}}^{n}$ and $K$ be a star body in ${\mathbb{R}}^{n}$ with $0\in{\rm int}(K)$. Then, for all ${\bf t}=(t_{1},\ldots,t_{s})\in{\mathbb{R}}^{s}$,

[TABLE]

where $c>0$ is an absolute constant. Equivalently, if ${\rm vol}_{n}(A_{j})={\rm vol}_{n}(K)$ for all $1\leqslant j\leqslant s$ then

[TABLE]

for all ${\bf t}\in{\mathbb{R}}^{s}$.

In the statement above, when $K$ is a star body with respect to [math] we use the notation $\|x\|_{K}$ for the gauge function of $K$, defined by $\inf\{r>0:x/r\in K\}$. The proof of Theorem 1.1 actually shows that one can have $c\geqslant c(n)/\sqrt{2}$, where $c(n)\to 1$ as $n\to\infty$. Gluskin and V. Milman use a symmetrization type result which is a consequence of the Brascamp-Lieb-Luttinger inequality: under the assumptions of Theorem 1.1 and the additional assumption that ${\rm vol}_{n}(A_{j})={\rm vol}_{n}(K)={\rm vol}_{n}(B_{2}^{n})$ for all $1\leqslant j\leqslant s$, one has

[TABLE]

for any ${\bf t}=(t_{1},\ldots,t_{s})\in{\mathbb{R}}^{s}$ and any $\alpha>0$.

Our starting point is a simple but useful identity; one has

[TABLE]

where $\nu_{{\bf t}}$ is the distribution of the random vector $\frac{1}{\|{\bf t}\|_{2}}(t_{1}X_{1}+\cdots+t_{s}X_{s})$ and $X_{j}$ are independent random vectors uniformly distributed on $C_{j}$. Starting with (1.2) we can actually give an alternative short proof of Theorem 1.1 in the case that we study.

Theorem 1.2**.**

Let ${\cal C}=(C_{1},\ldots,C_{s})$ be an $s$-tuple of centrally symmetric convex bodies and $K$ be a centrally symmetric convex body in $\mathbb{R}^{n}$ with ${\rm vol}_{n}(C_{j})={\rm vol}_{n}(K)=1$. Then, for any ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$,

[TABLE]

We are mainly interested in upper bounds for the quantity $\|{\bf t}\|_{C^{s},K}$. Since $\|{\bf t}\|_{C^{s},K}=\|{\bf t}\|_{(TC)^{s},TK}$ for any $T\in SL(n)$, we may restrict our attention to the case where $C$ is isotropic (see Section 2 for the definition and background information). In this case

[TABLE]

where $\mu_{{\bf t}}$ is an isotropic, compactly supported log-concave probability measure depending on ${\bf t}$ and, for any centered log-concave probability measure $\mu$ on ${\mathbb{R}}^{n}$,

[TABLE]

In order to get a feeling of what one would expect, let us note that if $\mu$ is an isotropic log-concave probability measure on ${\mathbb{R}}^{n}$ and $K$ is a centrally symmetric convex body of volume $1$ in ${\mathbb{R}}^{n}$ then

[TABLE]

where

[TABLE]

and $\nu,\sigma$ denote the Haar probability measures on $O(n)$ and $S^{n-1}$ respectively. It follows that

[TABLE]

Therefore, our goal is to obtain a constant of the order of $L_{C}\,\sqrt{n}M(K)$ in our upper estimate for $\|{\bf t}\|_{C^{s},K}$. Let us note here that the question to estimate the parameter $M(K)$ for an isotropic centrally symmetric convex body $K$ in ${\mathbb{R}}^{n}$, which will appear frequently in our upper bounds, remains open; one may hope that $L_{K}\,\sqrt{n}M(K)\leqslant c(\log n)^{b}$ for some absolute constant $b>0$. However, the currently best known estimate is

[TABLE]

This is proved in [15] (see also [16] for previous work on this question) and it is also shown that in the case where $K$ is a $\psi_{2}$-body with constant $\varrho$ one has

[TABLE]

We pass now to our bounds for $\|{\bf t}\|_{C^{s},K}$. Some straightforward upper and lower estimates are given in the next theorem.

Theorem 1.3**.**

Let $C$ be an isotropic convex body in $\mathbb{R}^{n}$ and $K$ be a centrally symmetric convex body in $\mathbb{R}^{n}$. Then, for any $s\geqslant 1$ and ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$,

[TABLE]

where $c_{1}>0$ is an absolute constant and $R(K^{\circ})$ is the radius of $K^{\circ}$.

A class of centrally symmetric convex bodies for which the upper bound of Theorem 1.3 can be applied is the class of $2$-convex bodies. More precisely, in Section 4.1 we see that if $K$ is an isotropic convex body in ${\mathbb{R}}^{n}$, which is also $2$-convex with constant $\alpha$, then

[TABLE]

for any isotropic centrally symmetric convex body $C$ and any ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$, where $c_{2}>0$ is an absolute constant. In particular, for any centrally symmetric convex body $K$ in ${\mathbb{R}}^{n}$ which is $2$-convex with constant $\alpha$ we have

[TABLE]

for all ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$, where $c_{3}>0$ is an absolute constant.

Starting again with (1.3) and using an argument which goes back to Bourgain (also, employing Paouris’ inequality and Talagrand’s comparison theorem) in Section 4.2 we obtain a general upper bound of different type.

Theorem 1.4**.**

Let $C$ be an isotropic convex body in $\mathbb{R}^{n}$ and $K$ be a centrally symmetric convex body in $\mathbb{R}^{n}$. Then,

[TABLE]

for every ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$, where $c>0$ is an absolute constant.

In the case where $C$ is a $\psi_{2}$-body with constant $\varrho$, a direct application of Talagrand’s theorem leads to a stronger estimate: If $C$ is an isotropic $\psi_{2}$-body with constant $\varrho$ and $K$ is a centrally symmetric convex body in $\mathbb{R}^{n}$ then

[TABLE]

for every ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$, where $c>0$ is an absolute constant.

Next, combining (1.3) with results of E. Milman from [24], we obtain some rather strong estimates in the case where $K$ has bounded cotype-$2$ constant (see Section 5). In the case $C=K$ we get:

Theorem 1.5**.**

Let $K$ be a centrally symmetric convex body in ${\mathbb{R}}^{n}$. For any $s\geqslant 1$ and any ${\bf t}=(t_{1},\ldots,t_{s})\in{\mathbb{R}}^{s}$ we have that

[TABLE]

where $C_{2}(X_{K})$ is the cotype-$2$ constant of the normed space $X_{K}$ with unit ball $K$, and $K_{{\rm iso}}$ is an isotropic image of $K$.

In Section 6 we consider the unconditional case; using an argument from [14] which is based on well-known results of Bobkov and Nazarov one has the following estimates.

Theorem 1.6**.**

If $K$ and $C_{1},\ldots,C_{s}$ are isotropic unconditional convex bodies in $\mathbb{R}^{n}$ then,

[TABLE]

for every ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$, where $c>0$ is an absolute constant.

As an application of Theorem 1.5 and of the “$\psi_{2}$-version” of Theorem 1.4 we can check that in the special case of the unit ball $B_{p}^{n}$ of $\ell_{p}^{n}$, $1\leqslant p\leqslant\infty$, one has the upper bound

[TABLE]

for every $s\geqslant 1$ and ${\bf t}\in{\mathbb{R}}^{s}$, where $c>0$ is an absolute constant (and, generally, $\overline{K}=\mathrm{vol}_{n}(K)^{-1/n}K$).

In Section 7 we discuss applications of the previous results to some randomized versions of vector balancing problems. Given two centrally symmetric convex bodies $C,K$ in $\mathbb{R}^{n}$, the parameter $\beta_{s}(C,K)$ is defined as follows:

[TABLE]

where $E_{2}^{s}:=\{-1,1\}^{s}$ is the discrete cube in ${\mathbb{R}}^{s}$. Given $x_{1},\ldots,x_{n}\in K$, by the triangle inequality it is clear that $\|\sum_{j=1}^{n}\epsilon_{j}x_{j}\|_{K}\leqslant n$ holds for every $\epsilon\in E_{2}^{n}$, thus $\beta_{n}(K,K)\leqslant n$. This bound is actually sharp: taking $K=B_{1}^{n}$ and $x_{j}=e_{j}$, the standard basis of $\mathbb{R}^{n}$, we get $\|\sum_{j=1}^{n}\epsilon_{j}e_{j}\|_{1}=n$ for every choice of signs. However, the upper bound for $\beta_{n}(K,K)$ can be significantly better for certain convex bodies, as suggested for example by a theorem of Spencer [28]: one has $\beta_{n}(B_{\infty}^{n},B_{\infty}^{n})\leqslant 6\sqrt{n}$.

We further define $\tilde{\beta}(C,K)=\sup_{k\geqslant n}\beta_{k}(C,K)$. Clearly, $\beta_{n}(C,K)\leqslant\tilde{\beta}(C,K)$. By a theorem of Bárány and Grinberg [3], one has $\tilde{\beta}(K,K)\leqslant 2n$. This result can also be derived by the trivial bound on $\beta_{n}(K,K)$ mentioned earlier and the general observation that

[TABLE]

A related result is the Dvoretzky-Hanani lemma (see for example [18, Lemma 2.2.1]) which asserts that for every centrally symmetric convex body $K$ in $\mathbb{R}^{n}$, for any $s\geqslant 1$ and any $x_{1},\ldots,x_{s}\in K$, there exist $\epsilon_{1},\ldots,\epsilon_{s}\in\{-1,1\}$ such that $\max_{k\leqslant s}\|\sum_{j=1}^{k}\epsilon_{j}x_{j}\|_{K}\leqslant 2n$.

The question that we discuss is whether one can achieve something better than the $O(n)$ bound for a random $s$-tuple $(x_{1},\ldots,x_{s})$ from $C$. In order to make this question precise, for any $\delta\in(0,1)$ we introduce the parameter

[TABLE]

The results of Section 4 and Section 5 allow us to obtain significantly better bounds for $\beta_{\delta,s}^{(R)}(C,K)$. In the statement below we restrict ourselves to the case $C=K$ and $s=n$; the reader may deduce analogous bounds for an arbitrary choice of $C$ or $s$.

Theorem 1.7**.**

Let $K$ be a centrally symmetric convex body in ${\mathbb{R}}^{n}$. Then, for any $\delta\in(0,1)$,

[TABLE]

where $c>0$ is an absolute constant and $K_{{\rm iso}}$ is an isotropic image of $K$. If $K$ is a $\psi_{2}$-body with constant $\varrho$ then

[TABLE]

Analogous results hold for $2$-convex bodies with constant $\alpha$, in which case we have

[TABLE]

or bodies with bounded cotype-$2$ constant; in this case we have

[TABLE]

In fact, the proof of Theorem 1.7 shows that the same upper bounds hold for the parameter $\kappa_{\delta,s}^{(R)}(C,K)$ which is defined as the smallest $r>0$ with the property that

[TABLE]

Note that, by definition, $\kappa_{\delta,s}^{(R)}(C,K)\geqslant\beta_{\delta,s}^{(R)}(C,K)$.

Finally, combining our approach with some classical results from asymptotic convex geometry we obtain variants of the main results of [12] as well as their dual estimates. We close this introductory section with the statements in the particular case $C=\overline{B_{2}^{n}}$.

Theorem 1.8**.**

Let ${\bf t}\in{\mathbb{R}}^{s}$. For any centrally symmetric convex body $K$ in ${\mathbb{R}}^{n}$ and any $S\subseteq E_{2}^{s}$ with $|S|\leqslant e^{cd(K)}$ we have

[TABLE]

and

[TABLE]

where $c_{i}>0$ are absolute constants.

The quantities $k(K)$ and $d(K)$ are well-known parameters of a centrally symmetric convex body $K$ which are introduced in Section 7; $k(K)=n(M(K)/b(K))^{2}$ is the Dvoretzky dimension of $K$ and

[TABLE]

where $m(K)\approx\sqrt{n}M(K)$ is the median (the Lévy mean) of $\|\cdot\|_{K}$ with respect to the standard Gaussian measure $\gamma_{n}$ on $\mathbb{R}^{n}$.

2 Backgound information and preliminary observations

In this section we introduce notation and terminology that we use throughout this work, and provide background information on isotropic convex bodies. We write $\langle\cdot,\cdot\rangle$ for the standard inner product in ${\mathbb{R}}^{n}$ and denote the Euclidean norm by $\|\cdot\|_{2}$. In what follows, $B_{2}^{n}$ is the Euclidean unit ball, $S^{n-1}$ is the unit sphere, and $\sigma$ is the rotationally invariant probability measure on $S^{n-1}$. Lebesgue measure in ${\mathbb{R}}^{n}$ is denoted by ${\rm vol}_{n}$. The letters $c,c^{\prime},c_{j},c_{j}^{\prime}$ etc. denote absolute positive constants whose value may change from line to line. Sometimes we might even relax our notation: $a\lesssim b$ will then mean “$a\leqslant cb$ for some (suitable) absolute constant $c>0$”, and $a\approx b$ will stand for “$a\lesssim b\land a\gtrsim b$”. If $A,B$ are sets, $A\approx B$ will similarly state that $c_{1}A\subseteq B\subseteq c_{2}A$ for some absolute constants $c_{1},c_{2}>0$.

A convex body in ${\mathbb{R}}^{n}$ is a compact convex set $C\subset{\mathbb{R}}^{n}$ with non-empty interior. We say that $C$ is centrally symmetric if $-C=C$. We say that $C$ is unconditional with respect to the standard orthonormal basis $\{e_{1},\ldots,e_{n}\}$ of ${\mathbb{R}}^{n}$ if $x=(x_{1},\ldots,x_{n})\in C$ implies that $(\epsilon_{1}x_{1},\ldots,\epsilon_{n}x_{n})\in C$ for any choice of signs $\epsilon_{j}\in\{-1,1\}$. The volume radius of $C$ is the quantity ${\rm vrad}(C)=\left(\mathrm{vol}_{n}(C)/\mathrm{vol}_{n}(B_{2}^{n})\right)^{1/n}$. Integration in polar coordinates shows that if the origin is an interior point of $C$ then the volume radius of $C$ can be expressed as

[TABLE]

where $\|\xi\|_{C}=\inf\{t>0:\xi\in tC\}$. We also consider the parameter

[TABLE]

The support function of $C$ is defined by h_{C}(y):=\max\bigl{\{}\langle x,y\rangle:x\in C\bigr{\}}, and the mean width of $C$ is the average

[TABLE]

of $h_{C}$ on $S^{n-1}$. The radius $R(C)$ of a centrally symmetric convex body $C$ is the smallest $R>0$ such that $C\subseteq RB_{2}^{n}$. We shall use the fact that $R(C)\leqslant c\sqrt{n}w(C)$; equivalently, $b(C)\leqslant c\sqrt{n}M(C)$, where $b(C)$ is the smallest $b>0$ with the property that $\|x\|_{C}\leqslant b\|x\|_{2}$ for all $x\in{\mathbb{R}}^{n}$. For notational convenience we write $\overline{C}$ for the homothetic image of volume $1$ of a convex body $C\subseteq\mathbb{R}^{n}$, i.e. $\overline{C}:=\mathrm{vol}_{n}(C)^{-1/n}C$.

The polar body $C^{\circ}$ of a centrally symmetric convex body $C$ in ${\mathbb{R}}^{n}$ is defined by

[TABLE]

The Blaschke-Santaló inequality states that ${\rm vol}_{n}(C){\rm vol}_{n}(C^{\circ})\leqslant{\rm vol}_{n}(B_{2}^{n})^{2}$, with equality if and only if $C$ is an ellipsoid. The reverse Santaló inequality of Bourgain and V. Milman [10] asserts that there exists an absolute constant $c>0$ such that, conversely,

[TABLE]

A convex body $C$ in ${\mathbb{R}}^{n}$ is called isotropic if it has volume $1$, it is centered, i.e. its barycenter is at the origin, and its inertia matrix is a multiple of the identity matrix: there exists a constant $L_{C}>0$ such that

[TABLE]

for all $\xi\in S^{n-1}$. We shall use the fact that if $C$ is isotropic then $R(C)\leqslant cnL_{C}$ for some absolute constant $c>0$. The hyperplane conjecture asks if there exists an absolute constant $A>0$ such that

[TABLE]

for all $n\geqslant 1$. Bourgain proved in [8] that $L_{n}\leqslant c\sqrt[4]{n}\log\!n$; later, Klartag [19] improved this bound to $L_{n}\leqslant c\sqrt[4]{n}$.

A Borel measure $\mu$ on $\mathbb{R}^{n}$ is called $\log$-concave if $\mu(\lambda A+(1-\lambda)B)\geqslant\mu(A)^{\lambda}\mu(B)^{1-\lambda}$ for any compact subsets $A$ and $B$ of ${\mathbb{R}}^{n}$ and any $\lambda\in(0,1)$. A function $f:\mathbb{R}^{n}\rightarrow[0,\infty)$ is called $\log$-concave if its support $\{f>0\}$ is a convex set and the restriction of $\log{f}$ to it is concave. It is known that if a probability measure $\mu$ is log-concave and $\mu(H)<1$ for every hyperplane $H$, then $\mu$ has a log-concave density $f_{{\mu}}$. Note that if $C$ is a convex body in $\mathbb{R}^{n}$ then the Brunn-Minkowski inequality implies that $\mathbf{1}_{C}$ is the density of a $\log$-concave measure.

If $\mu$ is a log-concave measure on ${\mathbb{R}}^{n}$ with density $f_{\mu}$, we define the isotropic constant of $\mu$ by

[TABLE]

where $\textrm{Cov}(\mu)$ is the covariance matrix of $\mu$ with entries

[TABLE]

We say that a $\log$-concave probability measure $\mu$ on ${\mathbb{R}}^{n}$ is isotropic if it is centered, i.e. if

[TABLE]

for all $\xi\in S^{n-1}$, and $\textrm{Cov}(\mu)$ is the identity matrix.

If $C$ is a centered convex body of volume $1$ in $\mathbb{R}^{n}$ then we say that a direction $\xi\in S^{n-1}$ is a $\psi_{\alpha}$-direction (where $1\leqslant\alpha\leqslant 2$) for $C$ with constant $\varrho>0$ if

[TABLE]

where

[TABLE]

From Markov’s inequality it is clear that if $C$ satisfies a $\psi_{\alpha}$-estimate with constant $\varrho$ in the direction of $\xi$ then for all $t\geqslant 1$ we have $\mathrm{vol}_{n}(\{x\in C:|\langle x,\xi\rangle|\geqslant t\|\langle\cdot,\xi\rangle\|_{L_{2}(C)}\})\leqslant 2e^{-t^{a}/\varrho^{\alpha}}$. Conversely, it is a standard fact that tail estimates of this form imply that $\xi$ is a $\psi_{\alpha}$-direction for $C$. Similar definitions may be given in the context of a centered log-concave probability measure $\mu$ on ${\mathbb{R}}^{n}$. From log-concavity it follows that every $\xi\in S^{n-1}$ is a $\psi_{1}$-direction for any $C$ or $\mu$ with an absolute constant $\varrho$: there exists $\varrho>0$ such that

[TABLE]

for all $n\geqslant 1$, all centered log-concave probability measures $\mu$ on ${\mathbb{R}}^{n}$ and all $\xi\in S^{n-1}$. We refer the reader to the book [11] for an updated exposition of isotropic log-concave measures and more information on the hyperplane conjecture.

We close this introductory section with a lemma that may be viewed as a form of generalization of Khinchine’s inequality, where the randomness is no longer that of Bernoulli $\{-1,1\}$ random variables but here is given by random vectors in the bodies $C_{1},\ldots,C_{s}$.

Lemma 2.1**.**

Let $C_{1},\ldots,C_{s}$ be convex bodies of volume $1$ and $K$ be a centrally symmetric convex body in $\mathbb{R}^{n}$. Then,

[TABLE]

where $c>0$ is an absolute constant.

The lemma follows immediately from the fact (see [11, Theorem 2.4.6]) that if $\mu$ is a log-concave probability measure on ${\mathbb{R}}^{k}$ and $f:{\mathbb{R}}^{k}\to{\mathbb{R}}$ is a seminorm then, for any $q\geqslant 1$,

[TABLE]

where $c>0$ is an absolute constant. We apply this fact on ${\mathbb{R}}^{ns}$ for the semi-norm

[TABLE]

and the uniform measure on $C_{1}\times\cdots\times C_{s}$.

3 A basic identity and a proof of the lower bound

In this section we assume that $C_{1},\ldots,C_{s}$ are centrally symmetric convex bodies of volume $1$ in ${\mathbb{R}}^{n}$ and study the quantity

[TABLE]

where ${\bf t}=(t_{1},\ldots,t_{s})$ and $K$ is a centrally symmetric convex body in ${\mathbb{R}}^{n}$. By the symmetry of the $C_{j}$’s we have that

[TABLE]

for all $\epsilon=(\epsilon_{1},\ldots,\epsilon_{s})\in E_{2}^{s}$, therefore we may always assume that $t_{1},\ldots,t_{s}\geqslant 0$. Our starting point is the next observation.

Lemma 3.1**.**

Let $X_{1},\ldots,X_{s}$ be independent random vectors, uniformly distributed on $C_{1},\ldots,C_{s}$ respectively. Given ${\bf t}=(t_{1}\ldots,t_{s})\in{\mathbb{R}}^{s}$, we write $\nu_{{\bf t}}$ for the distribution of the random vector $t_{1}X_{1}+\cdots+t_{s}X_{s}$. Then,

[TABLE]

Since $\|{\bf t}\|_{{\cal C},K}$ is a norm, we may always assume that $\|{\bf t}\|_{2}=1$. Note that $\nu_{{\bf t}}$ is an even log-concave probability measure on ${\mathbb{R}}^{n}$ (this is a consequence of the Prékopa-Leindler inequality; see [1]). We write $g_{{\bf t}}$ for the density of $\nu_{{\bf t}}$. The next lemma provides an upper bound for $\|g_{{\bf t}}\|_{\infty}=g_{{\bf t}}(0)$.

Lemma 3.2**.**

If $\|{\bf t}\|_{2}=1$ then $\|g_{{\bf t}}\|_{\infty}\leqslant e^{n}$.

Proof.

The proof employs a result of Bobkov and Madiman from [5] and the Shannon-Stam inequality (see [29]). Recall that the entropy functional of a random vector $X$ in ${\mathbb{R}}^{n}$ with density $g(x)$ is defined by

[TABLE]

provided the integral exists. Bobkov and Madiman have shown that if $g$ is log-concave then

[TABLE]

(the assumption that $g$ is log-concave is needed only for the right hand side inequality). Let ${\bf t}\in{\mathbb{R}}^{s}$ with $\|{\bf t}\|_{2}=1$ and $t_{1},\ldots,t_{s}\geqslant 0$. Then, if $X_{1},\ldots,X_{s}$ are independent random vectors with densities $g_{1},\ldots,g_{s}$ we have that

[TABLE]

This is an equivalent form of the Shannon-Stam inequality (see [22] and [13]). Since the density $g_{{\bf t}}$ of $t_{1}X_{1}+\cdots+t_{s}X_{s}$ is also log-concave, we may write

[TABLE]

which implies that

[TABLE]

In our case, $g_{j}={\bf 1}_{C_{j}}$, therefore $\|g_{j}\|_{\infty}=1$ and the lemma follows. ∎

The next lemma is an immediate consequence of [9, Lemma 2.3] (see also [25, Lemma 2.1]).

Lemma 3.3**.**

Let $f$ be a bounded positive density of a probability measure $\mu$ on ${\mathbb{R}}^{n}$. For any centrally symmetric convex body $K$ in ${\mathbb{R}}^{n}$ and any $p>0$ one has

[TABLE]

We apply Lemma 3.3 for the log-concave probability measure $\nu_{{\bf t}}$. For any ${\bf t}\in{\mathbb{R}}^{s}$ with $\|{\bf t}\|_{2}=1$ we have $\|g_{{\bf t}}\|_{\infty}=g_{{\bf t}}(0)\leqslant e^{n}$, therefore

[TABLE]

Combining this inequality with Lemma 3.1 we see that if ${\cal C}=(C_{1},\ldots,C_{s})$ is an $s$-tuple of centrally symmetric convex bodies of volume $1$ and $K$ is a centrally symmetric convex body in $\mathbb{R}^{n}$ then, for any $s\geqslant 1$ and any ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$

[TABLE]

This proves Theorem 1.2.

4 Upper bounds

In this section we assume that $C$ is an isotropic convex body in ${\mathbb{R}}^{n}$. We shall further exploit the identity of Lemma 3.1 to give upper estimates for $\|{\bf t}\|_{C^{s},K}$, where $K$ is a centrally symmetric convex body in ${\mathbb{R}}^{n}$.

As in the previous section, let $X_{1},\ldots,X_{s}$ be independent random vectors, uniformly distributed on $C$. Given ${\bf t}=(t_{1}\ldots,t_{s})\in{\mathbb{R}}^{s}$ with $\|{\bf t}\|_{2}=1$, we write $\nu_{{\bf t}}$ for the distribution of the random vector $t_{1}X_{1}+\cdots+t_{s}X_{s}$. It is then easily verified that the covariance matrix ${\rm Cov}(\nu_{{\bf t}})$ of $\nu_{{\bf t}}$ is a multiple of the identity: more precisely,

[TABLE]

It follows that if $g_{{\bf t}}$ is the density of $\nu_{{\bf t}}$ then $f_{{\bf t}}(x)=L_{C}^{n}g_{{\bf t}}(L_{C}x)$ is the density of an isotropic log-concave probability measure on ${\mathbb{R}}^{n}$. Indeed, we have

[TABLE]

for all $1\leqslant i,j\leqslant n$. From Lemma 3.2 we see that

[TABLE]

for all ${\bf t}\in{\mathbb{R}}^{s}$ with $\|{\bf t}\|_{2}=1$. We also have

[TABLE]

Definition 4.1**.**

Let $\mu$ be a centered log-concave probability measure on ${\mathbb{R}}^{n}$. For any star body $K$ in ${\mathbb{R}}^{n}$ we define

[TABLE]

With this definition, we can write

[TABLE]

for all ${\bf t}\in{\mathbb{R}}^{s}$ with $\|{\bf t}\|_{2}=1$. Then, our aim is to establish an upper bound for $I_{1}(\mu_{{\bf t}},K)$.

4.1 Simple upper and lower bounds

A first upper bound for $I_{1}(\mu_{{\bf t}},K)$ can be obtained if we use the simple inequality $\|y\|_{K}\leqslant b\|y\|_{2}$, where $b=b(K)=R(K^{\circ})$. We observe that

[TABLE]

because the last integral is bounded by $\sqrt{n}$: this follows immediately from the Cauchy-Schwarz inequality and the isotropicity of $\mu_{{\bf t}}$. On the other hand,

[TABLE]

where in the second inequality we are using [11, Theorem 2.4.6]. Inserting these two bounds into (4.1) we have the next theorem.

Theorem 4.2**.**

Let $C$ be an isotropic convex body in $\mathbb{R}^{n}$ and $K$ be a centrally symmetric convex body in $\mathbb{R}^{n}$. Then, for any $s\geqslant 1$ and ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$,

[TABLE]

where $c_{1}>0$ is an absolute constant.

There are some classes of centrally symmetric convex bodies that behave well with respect to the upper bound of Theorem 4.2. We discuss one of them in the next subsection.

4.2 $2$-convex bodies

Recall that if $K$ is a centrally symmetric convex body in $\mathbb{R}^{n}$ then the modulus of convexity of $K$ is the function $\delta_{K}:(0,2]\to{\mathbb{R}}$ defined by

[TABLE]

Then, $K$ is called $2$-convex with constant $\alpha$ if, for every $\varepsilon\in(0,2]$,

[TABLE]

Examples of $2$-convex bodies are given by the unit balls of subspaces of $L_{p}$-spaces, $1<p\leqslant 2$; one can check that the definition is satisfied with $\alpha\approx p-1$. Klartag and E. Milman have proved in [20] that if $K$ is a centrally symmetric convex body of volume $1$ in ${\mathbb{R}}^{n}$, which is also $2$-convex with constant $\alpha$, then

[TABLE]

where $c_{1}>0$ is an absolute constant. Moreover, if $K$ is isotropic then

[TABLE]

for an absolute constant $c_{2}>0$ (see, again, [20]). From Theorem 4.2 we immediately get the next estimate.

Theorem 4.3**.**

Let $C$ be an isotropic convex body in $\mathbb{R}^{n}$ and $K$ be an isotropic centrally symmetric convex body in ${\mathbb{R}}^{n}$ which is also $2$-convex with constant $\alpha$. Then for any $s\geqslant 1$ and ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$,

[TABLE]

where $c>0$ is an absolute constant. In particular, for any centrally symmetric convex body $K$ in ${\mathbb{R}}^{n}$ which is $2$-convex with constant $\alpha$, we have that

[TABLE]

Proof.

The first claim follows from the fact that $R(K^{\circ})\leqslant c_{2}^{-1}/(\sqrt{\alpha}\sqrt{n})$. For the second assertion we use the observation that $\mathbb{E}_{K^{s}}\left\|\sum_{j=1}^{s}t_{j}x_{j}\right\|_{K}=\mathbb{E}_{(TK)^{s}}\left\|\sum_{j=1}^{s}t_{j}x_{j}\right\|_{TK}$ for any $T\in SL(n)$, and hence we may assume that $K$ is isotropic. Since $L_{K}\leqslant c_{1}/\sqrt{\alpha}$ we see that

[TABLE]

where $c_{3}=c_{2}^{-1}c_{1}$.∎

4.3 A general upper bound

In this subsection we prove Theorem 1.4. By homogeneity it is enough to consider the case $\|{\bf t}\|_{2}=1$. Our starting point will be again (4.1). We have

[TABLE]

and hence our aim is to establish an upper bound for $I_{1}(\mu_{{\bf t}},K)$. We shall use a well-known inequality of Paouris from [26].

Theorem 4.4** (Paouris).**

If $\mu$ is an isotropic log-concave probability measure on $\mathbb{R}^{n}$, then

[TABLE]

for every $r\geqslant 1$, where $c_{1}>0$ is an absolute constant.

Note also that, since $R(C)\leqslant c_{2}nL_{C}$ and ${\rm supp}(\nu_{{\bf t}})\subseteq sC$, we have that

[TABLE]

for any ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$ with $\|{\bf t}\|_{2}=1$. Therefore, if we fix $r\geqslant 1$ and set $C_{{\bf t}}(r)={\rm supp}(\mu_{{\bf t}})\cap c_{1}r\sqrt{n}B_{2}^{n}$, we may write

[TABLE]

Turning our attention to the first term, we consider the log-concave probability measure $\mu_{{\bf t},r}$ with density

[TABLE]

and the stochastic process $(w_{y})_{y\in K^{\circ}}$ on $({\mathbb{R}}^{n},\mu_{{\bf t},r})$, where $w_{y}(x)=\langle x,y\rangle$. We also consider a standard Gaussian random vector $G$ in $\mathbb{R}^{n}$, and for $y\in K^{\circ}$ set $h_{y}(G)=\langle G,y\rangle$. Note that (see e.g. [1, Lemma 9.1.3])

[TABLE]

To bound $\mathbb{E}(\max_{y\in K^{\circ}}w_{y})$, we will use Talagrand’s comparison theorem (see [30]).

Theorem 4.5** (Talagrand’s comparison theorem).**

If $(Y_{t})_{t\in T}$ is a Gaussian process and $(X_{t})_{t\in T}$ is a stochastic process such that

[TABLE]

for some $\alpha>0$ and every $s,t\in T$, then

[TABLE]

It is easily checked that $\|h_{y}-h_{z}\|_{2}=\|y-z\|_{2}$ for all $y,z\in K^{\circ}$. To bound the $\psi_{2}$ norm of $w_{y}-w_{z}$, we use the inequality $\|h\|_{\psi_{2}}\leqslant\sqrt{\|h\|_{\psi_{1}}\|h\|_{\infty}}$. Note that

[TABLE]

and we also have

[TABLE]

for some absolute constant $c_{3}>0$ (here we also use the fact that $\mu(C_{{\bf t}}(r))\geqslant 1-e^{-r\sqrt{n}}\geqslant 1/2$). It follows that

[TABLE]

Theorem 4.5 then implies that

[TABLE]

Finally,

[TABLE]

Since $b(K)\leqslant c_{6}\sqrt{n}M(K)$ we have that

[TABLE]

if we choose

[TABLE]

Therefore,

[TABLE]

as claimed. $\hfill\Box$

Adapting the proof of Theorem 1.4 we can show that if $C$ is assumed a $\psi_{2}$-body with constant $\varrho$, which means that every direction $\xi$ is a $\psi_{2}$-direction for $C$ with constant $\varrho$, then a much better estimate is available.

Theorem 4.6**.**

Let $C$ be an isotropic convex body in $\mathbb{R}^{n}$, which is a $\psi_{2}$-body with constant $\varrho$, and $K$ be a centrally symmetric convex body in $\mathbb{R}^{n}$. Then for any $s\geqslant 1$ and every ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$,

[TABLE]

where $c>0$ is an absolute constant.

Proof.

We consider the Gaussian process $h_{y}(G)=\langle G,y\rangle$, where $G$ is a standard Gaussian random vector in $\mathbb{R}^{n}$, and recall that $\|h_{y}-h_{z}\|_{2}=\|y-z\|_{2}$ and

[TABLE]

The main observation is that if $\|{\bf t}\|_{2}=1$ then $\mu_{{\bf t}}$ is a $\psi_{2}$-measure with constant $\varrho$. Indeed, for any $\xi\in S^{n-1}$ we have (see [31, Proposition 2.6.1]) that if $w_{\xi}(x)=\langle x,\xi\rangle$ then

[TABLE]

and hence, for any $y,z\in K^{\circ}$, the $\psi_{2}$ norm of $w_{y}-w_{z}$ can be directly estimated as follows:

[TABLE]

Then, Theorem 4.5 and the fact that $L_{C}\leqslant c_{2}\varrho$ (see [11, Chapter 7]) imply that

[TABLE]

as claimed. ∎

5 Bodies with bounded cotype-$2$ constant

Let $K$ be a centrally symmetric convex body in ${\mathbb{R}}^{n}$. Recall that if $X_{K}$ is the normed space with unit ball $K$, we write $C_{2,k}(X_{K})$ for the best constant $C>0$ such that

[TABLE]

for all $x_{1},\ldots,x_{k}\in X$. Then, the cotype-2 constant of $X_{K}$ is defined as $C_{2}(X_{K}):=\sup_{k}C_{2,k}(X_{K})$. Replacing the $\epsilon_{j}$’s by independent standard Gaussian random variables $g_{j}$ in the definition above, one may define the Gaussian cotype-$2$ constant $\alpha_{2}(X_{K})$ of $X_{K}$. One can check that $\alpha_{2}(X_{K})\leqslant C_{2}(X_{K})$. E. Milman has proved in [24] that if $\mu$ is a finite, compactly supported isotropic measure on $\mathbb{R}^{n}$ then, for any centrally symmetric convex body $K$ in $\mathbb{R}^{n}$,

[TABLE]

Using (5.1) we can prove the following.

Theorem 5.1**.**

Let $C$ be an isotropic centrally symmetric convex body in $\mathbb{R}^{n}$ and $K$ be a centrally symmetric convex body in ${\mathbb{R}}^{n}$. Then for any $s\geqslant 1$ and ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$,

[TABLE]

where $c_{1},c_{2}>0$ are absolute constants. In particular, for any centrally symmetric convex body $K$ of volume $1$ in ${\mathbb{R}}^{n}$ we have that

[TABLE]

where $K_{{\rm iso}}$ is an isotropic image of $K$.

Proof.

Combining (5.1) with (4.1) we get

[TABLE]

for all ${\bf t}\in{\mathbb{R}}^{s}$ with $\|{\bf t}\|_{2}=1$. On the other hand, for any ${\bf t}\in{\mathbb{R}}^{s}$, by the symmetry of $C$ we have that

[TABLE]

where $c>0$ is an absolute constant (in the first inequality we are using the Kahane-Khintchine inequality and in the third inequality we are using [11, Theorem 2.4.6] for the semi-norm $(x_{1},\ldots,x_{s})\mapsto\left(\sum_{j=1}^{s}t_{j}^{2}\|x_{j}\|_{K}^{2}\right)^{1/2}$ on $C^{s}$, while in the last step we are using the Cauchy-Schwarz inequality for $\|\cdot\|_{K}$ on $C$). From Lemma 3.3 with $f={\bf 1}_{C}$ we see that

[TABLE]

and the result follows.

In the case $C=K$, we may assume that $K$ is isotropic and these bounds take the form

[TABLE]

This completes the proof. ∎

Remark 5.2**.**

Another interesting case is when $K$ has bounded type-$2$ constant. Recall that if $X_{K}$ is the normed space with unit ball $K$, we write $T_{2,k}(X_{K})$ for the best constant $T>0$ such that

[TABLE]

for all $x_{1},\ldots,x_{k}\in X$. Then, the type-2 constant of $X_{K}$ is defined as $T_{2}(X_{K}):=\sup_{k}T_{2,k}(X_{K})$. E. Milman has proved in [24] that if $\mu$ is a finite, compactly supported isotropic measure on $\mathbb{R}^{n}$ then, for any centrally symmetric convex body $K$ in $\mathbb{R}^{n}$,

[TABLE]

Using this inequality and following a similar argument, as in the cotype-$2$ case, we get:

Theorem 5.3**.**

Let $C$ be an isotropic centrally symmetric convex body in $\mathbb{R}^{n}$ and $K$ be a centrally symmetric convex body in ${\mathbb{R}}^{n}$. Then for any $s\geqslant 1$ and ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$,

[TABLE]

where $c_{1},c_{2}>0$ are absolute constants. In particular, for any centrally symmetric convex body $K$ of volume $1$ in ${\mathbb{R}}^{n}$ we have that

[TABLE]

Note that if $\mathrm{vol}_{n}(K)=1$ then $\sqrt{n}M(K)\geqslant c>0$, therefore the estimate is exact, up to the type-$2$ constant, and actually implies an upper bound for $L_{K}$.

6 The unconditional case

The case where $C_{1},\ldots,C_{s}$ and $K$ are isotropic unconditional convex bodies in ${\mathbb{R}}^{n}$ has been essentially studied in [14, Theorem 4.1].

Theorem 6.1**.**

There exists an absolute constant $c>0$ with the following property: if $K$ and $C_{1},\ldots,C_{s}$ are isotropic unconditional convex bodies in $\mathbb{R}^{n}$ then, for every $q\geqslant 1$,

[TABLE]

for every ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$. In particular,

[TABLE]

Proof.

We briefly sketch the argument, which is essentially the same as in [14]. We write $\mu_{n}$ for the uniform distribution on $B_{1}^{n}$, with density $\frac{d\mu_{n}(x)}{dx}=\frac{n!}{2^{n}}{\mathbf{1}}_{B_{1}^{n}}(x)$. If we set $\Delta_{n}=\{x\in{\mathbb{R}}_{+}^{n}:x_{1}+\cdots+x_{n}\leqslant 1\}$ then a simple computation shows that for every $n$-tuple of non-negative integers $p_{1},\ldots,p_{n}$, one has

[TABLE]

In [6] it is proved that for every isotropic unconditional convex body $K$ in ${\mathbb{R}}^{n}$ one has $cB_{\infty}^{n}\subseteq K\subseteq V_{n}$, where $V_{n}=\sqrt{3/2}nB_{1}^{n}$ and $c>0$ is an absolute constant. Therefore, $\|\cdot\|_{K}\leqslant c_{1}\|\cdot\|_{\infty}\leqslant c_{1}\|\cdot\|_{q}$, where $c_{1}>0$ is an absolute constant. We proceed to give an upper bound for

[TABLE]

where $q\geq 1$ is an integer. We write $x_{i}=(x_{i1},\ldots,x_{in})$ and define $y_{j}=(x_{1j},\ldots,x_{sj})$ for all $j=1,\ldots,n$. Then,

[TABLE]

Next, we apply a comparison theorem from [7]: for every function $F:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ which is centrally symmetric, coordinatewise increasing and absolutely continuous, we have that

[TABLE]

where $\mu_{A}$ is the uniform measure on the isotropic unconditional convex body $A$. It follows that

[TABLE]

where $c_{1}=\sqrt{3/2}$. Combining the above we see that

[TABLE]

Using the estimate $(n+2r)!\geq n!n^{2r}$ which holds for every $r\geq 0$, we get

[TABLE]

We now use another observation from [7]: if $q\geq 1$ is an integer and $P_{q}(y)=\sum_{q_{1}+\cdots+q_{s}=q}y_{1}^{q_{1}}\cdots y_{s}^{q_{s}}$, $y=(y_{1},\ldots,y_{s})\in{\mathbb{R}}_{+}^{s}$, then for any $y\in{\mathbb{R}}_{+}^{s}$ with $y_{1}+\cdots+y_{s}=1$ we have

[TABLE]

Applying this inequality to the $s$-tuple y=\frac{1}{\|{\bf t}\|_{2}^{2}}\big{(}t_{1}^{2},\ldots,t_{s}^{2}\big{)} we get

[TABLE]

Then, we easily conclude the proof.∎

Remark 6.2**.**

Using our approach we can obtain a similar upper bound directly. Consider ${\bf t}\in{\mathbb{R}}^{s}$ with $\|{\bf t}\|_{2}=1$. As usual, we have

[TABLE]

where $\mu_{{\bf t}}$ is an unconditional isotropic log-concave probability measure. Since $K$ is also unconditional and isotropic, we have $c_{1}B_{\infty}^{n}\subseteq K$ and hence $\|x\|_{K}\leqslant c_{1}^{-1}\|x\|_{\infty}$ for all $x\in{\mathbb{R}}^{n}$. Therefore,

[TABLE]

because $\mu_{{\bf t}}$ is an isotropic $\psi_{1}$-measure with an absolute constant $\varrho$ (see [1, Proposition 3.5.8]). Since $C$ is unconditional, we also have $L_{C}\leqslant c_{3}$ for some absolute constant $c_{3}>0$; it follows that

[TABLE]

for every ${\bf t}\in{\mathbb{R}}^{s}$. Of course the estimate of Theorem 6.1 is more delicate, and can be better by a $\sqrt{\log n}$-term, as it depends on the coordinates of ${\bf t}$.

Remark 6.3**.**

In [14] it is observed that the $\ell_{\infty}$-term in the estimate provided by Theorem 6.1 cannot be removed. If $C=\overline{B_{1}^{n}}$ and $K=\frac{1}{2}B_{\infty}^{n}$ then choosing the vector ${\bf e_{1}}=(1,0,0,\ldots,0)$ we have

[TABLE]

for some absolute constant $c>0$.

The example of the cube shows that the term $\sqrt{\log n}\|{\bf t}\|_{2}$ is necessary. Gluskin and V. Milman show in [17] that if $C=K=\frac{1}{2}B_{\infty}^{n}$ then

[TABLE]

where $u\approx\log n$ and $(t_{i}^{\ast})_{i\leqslant n}$ is the decreasing rearrangement of $(|t_{j}|)_{j=1}^{n}$. It is observed in [14, Remark 4.5] that this implies the lower bound

[TABLE]

Remark 6.4**.**

It is interesting to test the results of Section 4 and Section 5 on the example of the $\ell_{p}^{n}$-balls $B_{p}^{n}$. Let us first assume that $1\leqslant p\leqslant 2$. Then, $\ell_{p}^{n}$ has cotype-2 constant bounded by an absolute (independent from $p$ and $n$) constant. It is also known (see [1, Chapter 5]) that $M(B_{p}^{n})\approx n^{\frac{1}{p}-\frac{1}{2}}$ and $\mathrm{vol}_{n}(B_{p}^{n})^{1/n}\approx n^{-\frac{1}{p}}$. It follows that

[TABLE]

Since $\overline{B_{p}^{n}}$ is isotropic and its isotropic constant is also bounded by an absolute constant, Theorem 5.1 shows that

[TABLE]

for every $s\geqslant 1$ and ${\bf t}\in{\mathbb{R}}^{s}$, where $c_{1}>0$ is an absolute constant.

Next, let us assume that $2\leqslant q\leqslant\infty$. It is then known (see [1, Chapter 5]) that $\mathrm{vol}_{n}(B_{q}^{n})^{1/n}\approx n^{-\frac{1}{q}}$ and

[TABLE]

It follows that

[TABLE]

Since $\overline{B_{q}^{n}}$ is an isotropic $\psi_{2}$-convex body with constant $\varrho\approx 1$ (independent from $q$ and $n$ – see [4]) and its isotropic constant is also bounded by an absolute constant, Theorem 4.6 shows that

[TABLE]

for every $s\geqslant 1$ and ${\bf t}\in{\mathbb{R}}^{s}$, where $c_{2}>0$ is an absolute constant.

7 Applications to vector balancing problems

Let $\mu$ be an isotropic log-concave probability measure on ${\mathbb{R}}^{n}$ and $K$ be a centrally symmetric convex body in ${\mathbb{R}}^{n}$. Our starting observation is that

[TABLE]

Applying this fact for the measure $\mu_{{\bf t}}$, from (4.1) we immediately get the following.

Proposition 7.1**.**

Let $C$ be an isotropic convex body in $\mathbb{R}^{n}$ and $K$ be a centrally symmetric convex body in $\mathbb{R}^{n}$. For every ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$ there exists $U\in O(n)$ such that

[TABLE]

We know that if ${\rm vol}_{n}(K)=1$ then the quantity $\sqrt{n}M(K)$ is always greater than $c$. Therefore, Proposition 7.1 provides many examples in which the lower bound of Gluskin and V. Milman can be improved (note also the presence of $L_{C}$ in the right hand side of the inequality). For example, in the classical example of the cube $K=\dfrac{1}{2}B_{\infty}^{n}$ we have that $\sqrt{n}M(K)\approx\sqrt{\log n}$, which implies the following:

Corollary 7.2**.**

For every isotropic convex body $C$ in ${\mathbb{R}}^{n}$ and any ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$ there exists $U\in O(n)$ such that

[TABLE]

where $c>0$ is an absolute constant.

In this section we explore further this idea. We shall use a number of important facts from asymptotic convex geometry (see [11] for proofs and additional references). For any centrally symmetric convex body $K$ in ${\mathbb{R}}^{n}$ and any $q\neq 0$ we define

[TABLE]

Litvak, V. Milman and Schechtman have proved in [23] that

[TABLE]

for every $1\leqslant q\leqslant c_{1}k(K)$, where $c_{1}>0$ is an absolute constant and $k(K)=n(M(K)/b(K))^{2}$ is the Dvoretzky dimension of $K$. Moreover, Klartag and Vershynin have proved in [21] that

[TABLE]

for every $1\leqslant q\leqslant c_{2}d(K)$, where $d(K)\geqslant c_{3}k(K)$ is a parameter of $K$ defined by

[TABLE]

and $m(K)\approx\sqrt{n}M(K)$ is the median of $\|\cdot\|_{K}$ with respect to the standard Gaussian measure $\gamma_{n}$ on $\mathbb{R}^{n}$.

For any isotropic log-concave probability measure $\mu$ on $\mathbb{R}^{n}$ and any $q\neq 0$, $q>-n$, let

[TABLE]

Paouris has proved in [26] and [27] that

[TABLE]

for every $1\leqslant q\leqslant c_{4}q_{\ast}(\mu)$, where $q_{\ast}(\mu):=\max\{q:k(Z_{q}^{\circ}(\mu)\geqslant q\}$. It is known that $q_{\ast}(\mu)\geqslant c_{5}\sqrt{n}$. Moreover, if $\mu$ is a $\psi_{2}$-measure with constant $\varrho$ then $q_{\ast}(\mu)\geqslant c_{6}n/\varrho^{2}$.

Theorem 7.3**.**

Let $C$ be an isotropic centrally symmetric convex body in $\mathbb{R}^{n}$ and $K$ be a centrally symmetric convex body in $\mathbb{R}^{n}$. Then, for every ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$ and $S\subseteq E_{2}^{n}$ with $|S|\leqslant e^{q({\bf t})}$, a random $U\in O(n)$ satisfies

[TABLE]

with probability greater than $1-e^{-2q({\bf t})}$, where

[TABLE]

Proof.

We may assume that $\|{\bf t}\|_{2}=1$. We start by writing

[TABLE]

It follows that

[TABLE]

From Markov’s inequality, a random $U\in O(n)$ satisfies

[TABLE]

with probability greater than $1-e^{-2q({\bf t})}$. Since

[TABLE]

for every $\epsilon\in E_{2}^{s}$, we conclude that a random $U\in O(n)$ satisfies

[TABLE]

for all $\epsilon\in E_{2}^{s}$, with probability greater than $1-e^{-2q({\bf t})}$.

Next, fix any such $U$ and let $S\subseteq E_{2}^{n}$ with $|S|\leqslant e^{q({\bf t})}$. Using (7.3), (7.4) and Markov’s inequality, we see that a random $s$-tuple $(x_{1},\ldots,x_{s})\in C^{s}$ satisfies

[TABLE]

for all $\epsilon\in S$, with probability greater than $1-e^{-q({\bf t})}$.∎

Recall that if $C$ is a $\psi_{2}$-body with constant $\varrho$ then $\mu_{{\bf t}}$ is a $\psi_{2}$ isotropic log-concave probability measure with constant $\varrho$. In this case $q_{\ast}(\mu_{{\bf t}})\geqslant cn/\varrho^{2}$, and hence, in Theorem 7.3 we have $q({\bf t})\geqslant c\min\{n/\varrho^{2},d(K)\}$. Moreover, if $C=\overline{B_{2}^{n}}$ we have that $U(C)=\overline{B_{2}^{n}}$ for all $U\in O(n)$ and $\varrho\approx 1$. Therefore, we have the following corollary.

Corollary 7.4**.**

Let $C$ be an isotropic centrally symmetric convex body in $\mathbb{R}^{n}$ which is $\psi_{2}$ with constant $\varrho$, and $K$ be a centrally symmetric convex body in $\mathbb{R}^{n}$. Then, for every ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$ and $S\subseteq E_{2}^{s}$ with $|S|\leqslant e^{c\min\{n/\varrho^{2},d(K)\}}$, a random $U\in O(n)$ satisfies

[TABLE]

with probability greater than $1-e^{-c_{2}\min\{n/\varrho^{2},d(K)\}}$. In particular, for any centrally symmetric convex body $K$ in ${\mathbb{R}}^{n}$ and any $S\subseteq E_{2}^{s}$ with $|S|\leqslant e^{cd(K)}$ we have

[TABLE]

Remark 7.5**.**

Choosing $t_{1}=\cdots=t_{s}=1$, one may view the previous results as lower bounds for a “randomized” version of the parameter $\beta_{s}(C,K)$. A general lower bound for $\beta_{n}(C,K)$ was proved by Banaszczyk; in [2] he showed that if $C$ and $K$ are centrally symmetric convex bodies in ${\mathbb{R}}^{n}$ then

[TABLE]

for an absolute constant $c>0$. An alternative proof of this lower bound can be deduced from a more general result of Gluskin and V. Milman in [17]: If $\mathrm{vol}_{n}(K)=\mathrm{vol}_{n}(C)$ then, for any $0<u<1$ one has

[TABLE]

which implies that, for each ${\bf t}\in{\mathbb{R}}^{n}$, with probability greater than $1-e^{-n}$ with respect to $(x_{1},\ldots,x_{n})$ we have

[TABLE]

Banaszczyk’s theorem corresponds to the case $s=n$ and ${\bf t}=(1,1,\ldots,1)$. Starting from this observation, the first and third authors of this article proved in [12] several results in the spirit of Theorem 7.3 and Corollary 7.4. For example, they showed that if $K$ is a centrally symmetric convex body in $\mathbb{R}^{n}$ and $S\subseteq E_{2}^{n}$ then

[TABLE]

A concrete application of this fact is that, for every $1\leqslant p\leqslant\log n$ and any $S\subseteq E_{2}^{n}$ with $|S|\leqslant 2^{c_{p}n}$, a random $n$-tuple of points in $B_{2}^{n}$ satisfies, with probability greater than $1-e^{-n}$,

[TABLE]

for all $\epsilon=(\epsilon_{1},\ldots,\epsilon_{n})\in S$, while in the case $p>\log n$ one can deduce that for any $0<\delta<1$ and $S\subseteq E_{2}^{n}$ with $|S|\leqslant 2^{n^{1-\delta}}$, a random $n$-tuple of points in $B_{2}^{n}$ satisfies, with probability greater than $1-e^{-n}$,

[TABLE]

for all $\epsilon=(\epsilon_{1},\ldots,\epsilon_{n})\in S$. We can obtain (in fact, more direct) variants and generalizations of these bounds from Corollary 7.4 and the available information on $d(B_{p}^{n})$.

Following the proof of Theorem 7.3 we can also obtain upper bounds for the $\|\cdot\|_{K}$-norm of signed sums of random points from an isotropic body $C$.

Theorem 7.6**.**

Let $C$ be an isotropic centrally symmetric convex body in $\mathbb{R}^{n}$ and $K$ be a centrally symmetric convex body in $\mathbb{R}^{n}$. Then, for every ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$ and $S\subseteq E_{2}^{n}$ with $|S|\leqslant e^{p({\bf t})}$, a random $U\in O(n)$ satisfies

[TABLE]

with probability greater than $1-e^{-2p({\bf t})}$, where

[TABLE]

Proof.

We may assume that $\|{\bf t}\|_{2}=1$. We start by writing

[TABLE]

It follows that

[TABLE]

Then, we proceed as in the proof of Theorem 7.3 using Markov’s inequality, and then (7.2) and (7.4).∎

We can also obtain an analogue of Corollary 7.4 under the assumption that $C$ is a $\psi_{2}$-body with constant $\varrho$. In particular, we have:

Corollary 7.7**.**

Let $K$ be a centrally symmetric convex body in $\mathbb{R}^{n}$. Then, for every ${\bf t}=(t_{1},\ldots,t_{s})\in\mathbb{R}^{s}$ and any $S\subseteq E_{2}^{s}$ with $|S|\leqslant e^{ck(K)}$ we have

[TABLE]

Finally, we briefly describe the proof of Theorem 1.7. Recall that for any centrally symmetric convex body $K$ in ${\mathbb{R}}^{n}$ and any $\delta\in(0,1)$ the parameter $\beta_{\delta,s}^{(R)}(K,K)$ is defined by

[TABLE]

Proof of Theorem 1.7.

Our starting point is Lemma 2.1; applied for the vector ${\bf 1}=(1,\ldots,1)\in{\mathbb{R}}^{n}$, it shows that for any centrally symmetric convex body $K$ in $\mathbb{R}^{n}$,

[TABLE]

where $c>0$ is an absolute constant. On the other hand, by the symmetry of $K$ we have that, for any $q\geqslant 1$,

[TABLE]

Combining the above we have, in particular,

[TABLE]

It follows that a random $n$-tuple $(x_{1},\ldots,x_{n})\in K^{n}$ satisfies

[TABLE]

with probability greater than $1-e^{-q}$. Choosing $q=\log(2/\delta)$ we see that

[TABLE]

Inserting our upper bounds for $\|{\bf 1}\|_{K^{n},K}$ into (7.9) we conclude the proof. ∎

Acknowledgements. The contribution of the second named author to this work was made during a visit at the University of Missouri, Columbia; he would like to thank the Department of Mathematics for the warm hospitality. The third named author is supported by a PhD Scholarship from the Hellenic Foundation for Research and Innovation (ELIDEK); research number 70/3/14547.