A note on Mahler's conjecture

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.

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