A short elementary proof of reversed Brunn--Minkowski inequality for coconvex bodies

TL;DR

This paper presents a simple elementary proof that the volume function of coconvex bodies is strictly convex, leading to a reversed Brunn--Minkowski inequality, connecting classical convex geometry with coconvex bodies.

Contribution

It provides a straightforward proof of the reversed Brunn--Minkowski inequality for coconvex bodies based on classical volume concavity results.

Findings

Volume of coconvex bodies is strictly convex.

Reversed Brunn--Minkowski inequality follows from volume convexity.

Elementary proof simplifies understanding of coconvex volume properties.

Abstract

The theory of coconvex bodies was formalized by A.~Khovanski{\u\i} and V.~Timorin in \cite{KT}. It has fascinating relations with the classical theory of convex bodies, as well as applications to Lorentzian geometry. In a recent preprint \cite{schnei2}, R.~Schneider proved a result that implies a reversed Brunn--Minkowski inequality for coconvex bodies, with description of equality case. In this note we show that this latter result is an immediate consequence of a more general result, namely that the volume of coconvex bodies is strictly convex. This result itself follows from a classical elementary result about the concavity of the volume of convex bodies inscribed in the same cylinder.