Volume inequalities for the $i$-th-Convolution bodies

TL;DR

This paper extends classical inequalities to convolution bodies of convex sets, providing bounds on their volumes and revealing new geometric relations, especially for the limiting convolution bodies and their special cases.

Contribution

It introduces new volume bounds for convolution bodies, including a sharp inequality for the $(n-1)$-th limiting convolution body, extending Zhang's inequality.

Findings

Upper bounds for volume of sum of convex bodies

Lower bounds for volume of limiting convolution bodies

Sharp inequality for the $(n-1)$-th limiting convolution body

Abstract

We obtain a new extension of Rogers-Shephard inequality providing an upper bound for the volume of the sum of two convex bodies $K$ and $L$. We also give lower bounds for the volume of the $k$-th limiting convolution body of two convex bodies $K$ and $L$. Special attention is paid to the $(n-1)$-th limiting convolution body, for which a sharp inequality, which is equality only when $K=-L$ is a simplex, is given. Since the $n$-th limiting convolution body of $K$ and $-K$ is the polar projection body of $K$, these inequalities can be viewed as an extension of Zhang's inequality.