On Godbersen's Conjecture

TL;DR

This paper generalizes a geometric conjecture related to convex bodies and their negatives, connects it to Godbersen's conjecture on mixed volumes, and establishes new bounds using a functional inequality.

Contribution

It introduces a natural generalization of a geometric conjecture, links it to Godbersen's conjecture, and derives the best known upper bounds for certain mixed volumes.

Findings

Established a new upper bound for mixed volumes V(K[j], -K[n-j])

Proved a functional inequality generalizing Colesanti's difference function inequality

Connected a geometric conjecture to Godbersen's conjecture through volume inequalities

Abstract

We provide a natural generalization of a geometric conjecture of F\'{a}ry and R\'{e}dei 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.