On a multi-integral norm defined by weighted sums of log-concave random vectors

TL;DR

This paper establishes an upper bound for a multi-integral norm involving log-concave vectors, linking it to the isotropic constant and a geometric parameter of convex bodies, thus connecting the slicing problem to norm estimates.

Contribution

It introduces a new inequality bounding a multi-integral norm of log-concave vectors in terms of isotropic constants and geometric parameters, reducing a key question in convex geometry.

Findings

Bound on the multi-integral norm involving log-concave vectors.

Reduction of V. Milman's question to estimating M(K).

Application of slicing problem results to norm inequalities.

Abstract

Let $C$ and $K$ be centrally symmetric convex bodies in ${\mathbb R}^n$. We show that if $C$ is isotropic then \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 \leq c_1L_C(\log n)^5\,\sqrt{n}M(K)\|{\bf t}\|_2\end{equation*} for every $s\geq 1$ and ${\bf t}=(t_1,\ldots ,t_s)\in {\mathbb R}^s$, where $L_C$ is the isotropic constant of $C$ and $M(K):=\int_{S^{n-1}}\|\xi\|_Kd\sigma (\xi)$. This reduces a question of V.~Milman to the problem of estimating from above the parameter $M(K)$ of an isotropic convex body. The proof is based on an observation that combines results of Eldan, Lehec and Klartag on the slicing problem: If $\mu $ is an isotropic log-concave probability measure on ${\mathbb R}^n$ then, for any centrally symmetric convex body $K$ in ${\mathbb R}^n$ we have that $$I_1(\mu ,K):=\int_{{\mathbb R}^n}\|x\|_K\,d\mu(x)\leq c_2\sqrt{n}(\log n)^5\,M(K).$$ We illustrate the use of this inequality with further applications.