On Minkowski's Monotonicity Problem

More than 120 years ago, Minkowski published a seminal paper that laid the foundation for the field of convex geometry (as well as several other areas of mathematics). Despite numerous advances in the intervening years, there are fundamental questions that arise in Minkowski's original work that remain open to this day. Major progress on one of these questions, the characterization of the extremals of the Alexandrov-Fenchel inequality, was achieved only in the past few years in joint work with Yair Shenfeld. In this talk I aim to discuss partial progress, in joint work with Shouda Wang, on another open problem that dates back to the same work: the equality characterization for the monotonicity of mixed volumes. No prior background on convex geometry will be assumed; instead I will aim to explain the problem, from a geometric analysis perspective, as a mixed analogue of the classical fact (which dates back to Euler) that a surface with vanishing Gaussian curvature is a ruled surface.