Mahler-type volume inequality for convex bodies with tetrahedral symmetry

TL;DR

This paper proves a volume inequality involving convex bodies and their difference bodies, providing a new proof in 2D and establishing a tetrahedral symmetry case in 3D with equality conditions.

Contribution

It offers a new proof of a known 2D volume inequality and extends the inequality to convex bodies with tetrahedral symmetry in 3D, identifying equality cases.

Findings

New proof of the 2D volume inequality for convex bodies.

Extension of the inequality to 3D convex bodies with tetrahedral symmetry.

Identification of tetrahedron as the equality case in 3D.

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.