The Symmetric Mahler Inequality in Dimension Three via Admissible Shadow Systems

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}.