On a generalization of Godbersen's conjecture
Jan Kotrbatý
TL;DR
This work investigates Godbersen's conjecture, which refines the Rogers–Shephard inequality for a convex body's difference body, by proposing and proving a higher-order generalization involving D_pK. The authors establish the conjecture for anti-blocking convex bodies, linking the inequality to higher-rank mixed volumes and to the Alesker product of smooth, translation-invariant valuations, thereby embedding these convex-geometry inequalities in the algebraic framework of valuations. They show that the generalized conjecture implies Schneider's higher-order inequality, and they discuss symmetry, equality cases, and reductions to easier scenarios. Overall, the paper advances a valuation-theoretic perspective on higher-order difference-body inequalities, suggesting new avenues for unifying classical geometric inequalities via the Alesker algebra.
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.