Equality cases for the $L_p$-Rogers--Shephard inequality in the plane and for locally anti-blocking bodies in $\mathbb{R}^n$

Abstract: The classical Rogers--Shephard inequalities were extended to the Firey $L_p$-summation by Bianchini and Colesanti in the plane and by Zvavitch and the second and fourth authors for locally anti-blocking convex bodies in $\mathbb{R}^n$, leaving open the equality cases. We characterize the equality cases of these inequalities: in both cases, for $p>1$, equality holds if and only if the convex body is a simplex with one vertex at the origin.