On unbalanced difference bodies and Godbersen's conjecture

TL;DR

This paper investigates Godbersen's conjecture on mixed volumes of convex bodies, demonstrating that certain averaged inequalities hold universally and extending classical inequalities like Rogers-Shephard's to difference bodies.

Contribution

The paper proves that several geometric inequalities related to mixed volumes are valid on average for any convex body, and extends the Rogers-Shephard inequality to difference bodies.

Findings

Certain linear combinations of mixed volumes are bounded by binomial coefficients.

An average inequality shows Godbersen's conjecture holds 'on average' for any convex body.

Generalization of Rogers-Shephard inequality for difference bodies.

Abstract

The longstanding Godbersen's conjecture states that for any convex body $K \subset \mathbb R^n$ of volume $1$ and any $j \in \{0, \ldots, n\}$, the mixed volume $V_j = V(K[j], -K[n - j])$ is bounded by $\binom{n}{j}$, with equality if and only if $K$ is a simplex. We demonstrate that several consequences of this conjecture are true: certain families of linear combinations of the $V_j$, arising from different geometric constructions, are bounded above by their values when one substitutes $\binom{n}{j}$ for $V_j$, with equality if and only if $K$ is a simplex. One of our results implies that for any $K$ of volume $1$ we have $\frac{1}{n + 1} \sum_{j = 0}^n \binom{n}{j}^{-1} V_j \le 1$, showing that Godbersen's conjecture holds ``on average'' for any body. Another result generalizes the well-known Rogers-Shephard inequality for the difference body.