General Measure Extensions of Projection Bodies

TL;DR

This paper generalizes affine isoperimetric inequalities related to projection bodies to arbitrary measures and functions, extending classical results like Zhang's inequality, and applies these to the Gaussian measure.

Contribution

It introduces measure- and function-based extensions of projection bodies and associated inequalities, broadening the scope of classical convex geometric inequalities.

Findings

Generalized Zhang's inequality to arbitrary measures

Extended projection body operator to functions and measures

Applied results to the Gaussian measure

Abstract

The inequalities of Petty and Zhang are affine isoperimetric-type inequalities providing sharp bounds for $\text{vol}^{n-1}_{n}(K)\text{vol}_n(\Pi^\circ K),$ where $\Pi K$ is a projection body of a convex body $K$. In this paper, we present a number of generalizations of Zhang's inequality to the setting of arbitrary measures. In addition, we introduce extensions of the projection body operator $\Pi$ to the setting of arbitrary measures and functions, while providing associated inequalities for this operator; in particular, Zhang-type inequalities. Throughout, we apply shown results to the standard Gaussian measure.