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