Symmetric Mahler’s conjecture for the volume product in the 3-dimensional case

Abstract

We prove Mahler’s conjecture concerning the volume product of centrally symmetric, convex bodies in ${\mathbb{R}}^n$ in the case where $n=3$. More precisely, we show that, for every $3$-dimensional, centrally symmetric, convex body $K\subset {\mathbb{R}}^3$, the volume product $\left|K\right|\left|K^{\circ }\right|$ is greater than or equal to $32/3$ with equality if and only if $K$ or $K^{\circ }$ is a parallelepiped.