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

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 $p\in(1,\infty]$, the only extremizers are simplices with a vertex at the origin.