On Minkowski's monotonicity problem

TL;DR

This paper solves a longstanding problem in convex geometry by characterizing equality cases in Minkowski's monotonicity of mixed volumes, providing new insights into the structure of convex bodies and their geometric measures.

Contribution

It proves a key part of Schneider's conjecture for all convex bodies in any dimension and fully resolves it in three dimensions, advancing understanding of mixed volume support.

Findings

Resolved one direction of Schneider's conjecture for arbitrary convex bodies.

Fully proved Schneider's conjecture in three dimensions.

Connected geometric properties of convex bodies to classical surface curvature results.

Abstract

We address an old open question in convex geometry that dates back to the work of Minkowski: what are the equality cases of the monotonicity of mixed volumes? The problem is equivalent to that of providing a geometric characterization of the support of mixed area measures. A conjectural characterization was put forward by Schneider (1985), but has been verified to date only for special classes of convex bodies. In this paper we resolve one direction of Schneider's conjecture for arbitrary convex bodies in $\mathbb{R}^n$, and resolve the full conjecture in $\mathbb{R}^3$. Among the implications of these results is a mixed counterpart of the classical fact, due to Monge, Hartman--Nirenberg, and Pogorelov, that a surface with vanishing Gaussian curvature is a ruled surface.