A note on the reduction of the slicing problem to centrally symmetric convex bodies

TL;DR

This paper determines the optimal constant in an inequality linking the isotropic constant of convex bodies to a related convex body derived from the covariogram function, advancing the understanding of the hyperplane conjecture.

Contribution

It establishes the best possible constant in a key inequality connecting isotropic convex bodies and associated convex bodies, reducing the hyperplane conjecture to centrally symmetric cases.

Findings

Established the optimal constant C for the inequality.

Derived sharp inclusion results for convex bodies in a specific family.

Connected the isotropic constant to properties of the covariogram function.

Abstract

In this paper, we obtain the best possible value of the absolute constant $C$ such that for every isotropic convex body $K \subseteq \mathbb{R}^n$ the following inequality (which was proved by Klartag and reduces the hyperplane conjecture to centrally symmetric convex bodies) is satisfied: $$ L_K\leq CL_{K_{n+2}(g_K)}. $$ Here $L_K$ denotes the isotropic constant of $K$, $g_K$ its covariogram function, which is log-concave, and, for any log-concave function $g$, $K_{n+2}(g)$ is a convex body associated to the log-concave function $g$, which belongs to a uniparametric family introduced by Ball. In order to obtain this inequality, sharp inclusion results between the convex bodies in this family are obtained whenever $g$ satisfies a better type of concavity than the log-concavity, as $g_K$ is, indeed $\frac{1}{n}$-concave.