Empirical forms of the Petty projection inequality

TL;DR

This paper explores empirical versions of the Petty projection inequality in convex geometry, deriving sharp extremal inequalities for functionals of random convex sets from a stochastic perspective.

Contribution

It introduces new empirical forms of the Petty projection inequality and establishes sharp extremal inequalities for functionals of random convex sets.

Findings

Derived sharp extremal inequalities for mixed projection bodies.

Re-examined key geometric relationships from a stochastic perspective.

Connected projection bodies, centroid bodies, and mixed volume inequalities.

Abstract

The Petty projection inequality is a fundamental affine isoperimetric principle for convex sets. It has shaped several directions of research in convex geometry which forged new connections between projection bodies, centroid bodies, and mixed volume inequalities. We establish several different empirical forms of the Petty projection inequality by re-examining these key relationships from a stochastic perspective. In particular, we derive sharp extremal inequalities for several multiple-entry functionals of random convex sets, including mixed projection bodies and mixed volumes.