The Symmetric Mahler Inequality in Dimension Three via Admissible Shadow Systems

TL;DR

This paper presents a new purely geometric proof of the three-dimensional symmetric Mahler inequality using symmetric admissible shadow systems, extending techniques from previous work on the Mahler conjecture.

Contribution

It introduces symmetric admissible shadow systems as a novel geometric method to prove the Mahler inequality in three dimensions.

Findings

New geometric proof of the symmetric Mahler inequality in  dimensions.

Extension of shadow system techniques from non-symmetric to symmetric cases.

Simplification of previous proofs combining algebraic-topological and geometric methods.

Abstract

The three-dimensional symmetric Mahler inequality states that, for every origin-symmetric convex body \(K=-K\subset \mathbb{R}^3\), \[ \VP(K)= |K|\,|K^\circ|\geq \frac{32}{3}. \] It was recently proved by Iriyeh--Shibata \cite{IS2020}, and a shorter proof was later given by Fradelizi--Hubard--Meyer--Rold\'an-Pensado--Zvavitch \cite{FHMRZ}. Both proofs combine ingenious equipartition arguments of algebraic-topological origin with delicate geometric estimates inspired by Meyer's argument for unconditional bodies. In this paper, we give a new proof of this inequality using a purely geometric approach, based on what we call symmetric admissible shadow systems. This is a natural extension of the new techniques developed in our proof of the three-dimensional non-symmetric Mahler conjecture \cite{CLXX-Mahler}.