On a generalization of Godbersen's conjecture

Abstract

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.

arXiv comment10 pages, minor corrections, to appear in IMRN