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 $|K{|}^{n-1}|{\Pi }^{\ast }(K)|$ (where ${\Pi }^{\ast }(K)$ 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.

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