On a Generalization of Godbersen’s Conjecture

The long-standing Godbersen’s conjecture asserts that the Rogers–Shephard inequality for the volume of the difference body is refined by an inequality for the mixed volume of a convex body and its reflection about the origin. The conjecture is known in several special cases, notably for anti-blocking convex bodies. In this note, we propose a generalization of Godbersen’s conjecture that refines Schneider’s generalization of the Rogers–Shephard inequality to higher-order difference bodies and prove our conjecture for anti-blocking convex bodies. Moreover, we relate the conjectured inequality to the higher-rank mixed volume defined by the author and Wannerer, which leads to an equivalent formulation in terms of the Alesker product of smooth, translation invariant valuations.