Godbersen's conjecture and the L_p-Rogers-Shephard inequality

2026/07/22 by Jan Kotrbatý, Mohamed A. Mouamine
#math.MG

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Abstract

We prove that the mixed volume of a convex body with its reflection about the origin is maximized by simplices. This confirms a conjecture of C. Godbersen from 1938 and refines the Rogers-Shephard inequality. We also prove that, among convex polytopes, simplices are the only extremizers. Finally, we use this inequality to prove the L_p-version of the Rogers-Shephard inequality for convex bodies containing the origin and show that, for any p∈(1,∞], the only extremizers are simplices with a vertex at the origin.

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