Stochastic forms of Brunn's principle

TL;DR

This paper develops stochastic geometric formulations of Brunn's principle for convex sets and concave functions, introducing shadow systems and revisiting integral inequalities to extend the principle's applicability.

Contribution

It introduces a stochastic framework for concave functions and establishes local dimensional versions of Brunn's principle using shadow systems and integral inequalities.

Findings

Established local stochastic versions of Brunn's principle for convex sets.

Developed shadow systems of convex epigraphs and hypographs.

Revisited Rinott's approach to integral rearrangement inequalities.

Abstract

A number of geometric inequalities for convex sets arising from Brunn's concavity principle have recently been shown to yield local stochastic formulations. Comparatively, there has been much less progress towards stochastic forms of related functional inequalities. We work towards a stochastic geometry of concave functions to establish local versions of dimensional forms of Brunn's principle a la Borell, Brascamp-Lieb, and Rinott. To do so, we define shadow systems of convex epigraphs and hypographs, and revisit Rinott's approach in the context of multiple integral rearrangement inequalities.