A note on Mahler's conjecture

Abstract

Let $K$ be a convex body in ${\mathbb{R}}^n$ with Santaló point at 0\. We show that if $K$ has a point on the boundary with positive generalized Gauß curvature, then the volume product $|K||K^{\circ }|$ is not minimal. This means that a body with minimal volume product has Gauß curvature equal to 0 almost everywhere and thus suggests strongly that a minimal body is a polytope.