Volume inequalities and additive maps of convex bodies

TL;DR

This paper develops new inequalities related to volume and additive maps of convex and star bodies, generalizing classical Brunn-Minkowski results and introducing a novel inequality for polar projection bodies.

Contribution

It introduces analogues of classical inequalities for rotation intertwining additive maps of convex and star bodies, extending the scope of Brunn-Minkowski theory.

Findings

New inequalities for convex and star bodies

Generalizations of projection and intersection body results

A novel Brunn-Minkowski inequality for polar projection bodies

Abstract

Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary, a new Brunn-Minkowski inequality is obtained for the volume of polar projection bodies.