Equipartitions and Mahler volumes of symmetric convex bodies

TL;DR

This paper provides a simplified proof of the three-dimensional Mahler conjecture for symmetric convex bodies, introduces new equipartition and volume inequalities, and extends the class of convex sets known to satisfy the conjecture.

Contribution

It offers a concise, self-contained proof of key inequalities and the Mahler conjecture in 3D, along with new stability results and a broader family of convex bodies.

Findings

Proof of the three-dimensional Mahler conjecture for symmetric convex bodies.

Introduction of a refined equipartition theorem using equivariant topology.

Identification of a large family of convex sets satisfying the Mahler conjecture.

Abstract

Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture {\mf for symmetric convex bodies}. Our contributions include, in particular, simple self-contained proofs of their two key statements. The first of these is an equipartition (ham sandwich type) theorem which refines a celebrated result of Hadwiger and, as usual, can be proved using ideas from equivariant topology. The second is an inequality relating the product volume to areas of certain sections and their duals. We observe that these ideas give a large family of convex sets in every dimension for which the Mahler conjecture holds true. Finally we give an alternative proof of the characterization of convex bodies that achieve the equality case and establish a {\mf new} stability result.