On the volume of convolution bodies in the plane

TL;DR

This paper investigates the volume of convolution bodies in the plane, revealing that ellipsoids do not maximize this volume for fixed convex bodies, contrasting with known inequalities as delta approaches 1.

Contribution

It demonstrates that in two dimensions, ellipsoids are not volume maximizers for convolution bodies, challenging previous expectations based on the Petty projection inequality.

Findings

Ellipsoids do not maximize the volume of convolution bodies in the plane.

The behavior differs from the limit case as delta approaches 1.

Contradicts the Petty projection inequality in this context.

Abstract

For every convex body $K \subset \mathbb R^n$ and $\delta \in (0,1)$, the $\delta$-convolution body of $K$ is the set of $x \in \mathbb R^n$ for which $\left|K \cap (K+x)\right|_n \geq \delta \left|K\right|_n$. We show that for $n=2$ and any $\delta \in (0,1)$, ellipsoids do not maximize the volume of the $\delta$-convolution body of $K$, when $K$ runs over all convex bodies of a fixed volume. This behavior is somehow unexpected and contradicts the limit case $\delta \to 1^-$, which is governed by the Petty projection inequality.