Log-Concavity Properties of Minkowski Valuations

TL;DR

This paper establishes new Orlicz Brunn-Minkowski inequalities for Minkowski valuations, extending classical log-concavity properties and providing a unified approach through two refined methods.

Contribution

It introduces novel inequalities for Minkowski valuations, generalizes classical results, and offers a new classification of translation invariant valuations.

Findings

Established new Orlicz Brunn-Minkowski inequalities

Unified approach for classifying valuations with these properties

Extended classical log-concavity results

Abstract

New Orlicz Brunn-Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are explored. It is shown that both lead to the same class of Minkowski valuations for which these inequalities hold. An appendix by Semyon Alesker contains the proof of a new classification of generalized translation invariant valuations.