On Godbersen’s conjecture

Abstract

We provide a natural generalization of a geometric conjecture of Fary and Redei regarding the volume of the convex hull of \(K \subset {\mathbb {R}}^n\), and its negative image \(-K\). We show that it implies Godbersen’s conjecture regarding the mixed volumes of the convex bodies \(K\) and \(-K\). We then use the same type of reasoning to produce the currently best known upper bound for the mixed volumes \(V(K[j], -K[n-j])\), which is not far from Godbersen’s conjectured bound. To this end we prove a certain functional inequality generalizing Colesanti’s difference function inequality.