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

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