On a generalization of Godbersen's conjecture

TL;DR

This paper proposes a generalized form of Godbersen's conjecture related to mixed volumes of convex bodies, extends known cases, and connects it to higher-rank mixed volume and valuation theory.

Contribution

It introduces a new generalization of Godbersen's conjecture, proves it for anti-blocking convex bodies, and links it to advanced valuation concepts.

Findings

Proved the conjecture for anti-blocking convex bodies.

Established a connection to higher-rank mixed volume.

Reformulated the conjecture using Alesker product and valuation theory.

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.