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arXiv · 2605.13795

The Symmetric Mahler Inequality in Dimension Three via Admissible Shadow Systems

Shibing Chen·Yuanyuan Li·Dongmeng Xi·Zhefeng Xu
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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}.

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Shibing Chen, Yuanyuan Li, Dongmeng Xi, Zhefeng Xu. 2026-05-13. The Symmetric Mahler Inequality in Dimension Three via Admissible Shadow Systems. https://arxiv.org/abs/2605.13795

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