Zhang's inequality for log-concave functions

TL;DR

This paper extends Zhang's inequality from convex bodies to integrable log-concave functions, providing a new affine isoperimetric inequality and characterizing the cases of equality.

Contribution

It generalizes Zhang's inequality to log-concave functions and characterizes the equality cases in this broader setting.

Findings

Extended Zhang's inequality to log-concave functions

Characterized equality cases for the inequality

Provided new affine isoperimetric inequality for functions

Abstract

Zhang's reverse affine isoperimetric inequality states that among all convex bodies $K\subseteq\mathbb{R}^n$, the affine invariant quantity $|K|^{n-1}|\Pi^*(K)|$ (where $\Pi^*(K)$ denotes the polar projection body of $K$) is minimized if and only if $K$ is a simplex. In this paper we prove an extension of Zhang's inequality in the setting of integrable log-concave functions, characterizing also the equality cases.