A note on Mahler's conjecture

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Let K be a convex body in Rⁿ 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∘| 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.