On Minkowski's monotonicity problem

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.