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 $ \vert K\vert $, 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.