Mahler-type volume inequality for convex bodies with tetrahedral symmetry

Abstract

Let $K$ be a convex body in ${\mathbb{R}}^n$. We denote the volume of $K$ by $|K|$, and the polar body of its difference body $K-K$ by $(K-K{)}^{\circ }$. We provide a new proof of the well-known estimate \[ |K||(K - K)^{\circ}| \geq \frac{3}{2} \] for $K\subset {\mathbb{R}}^2$, with equality attained for a triangle. For $K\subset {\mathbb{R}}^3$ with tetrahedral symmetry, we prove that \[ |K| |(K - K)^{\circ}| \geq \frac{2}{3}, \] with equality attained for a tetrahedron.