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Shlomo Reisner, Carsten Schütt, Elisabeth M. Werner, Mahler's Conjecture and Curvature, International Mathematics Research Notices, Volume 2012, Issue 1, 2012, Pages 1–16, https://doi.org/10.1093/imrn/rnr003
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Abstract
Let K be a convex body in 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 a Gauß curvature equal to 0 almost everywhere and thus suggests strongly that a minimal body is a polytope.
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