Zhang’s Inequality for Log-Concave 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| Π∗(K)| (where Π∗(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.