Videos

Around the Alexandrov-Fenchel Inequality

Presenter
January 27, 2025
Abstract
In the late 1800s, in the course of his study of classical problems of number theory, the young Hermann Minkowski discovered the importance of a new kind of geometric object that we now call a convex set. He soon developed a rich theory for understanding such sets, laying the foundations of convex geometry that are widely used to this day. Among the most surprising observations of Minokwski's theory is that the classical isoperimetric theorem---which states that the ball has the smallest surface area among all bodies of a given volume---is just one special case of a much more general phenomenon. When one fixes geometric parameters other than volume, Minkowski discovered that the resulting "bubbles" can be strikingly bizarre. A complete understanding of such objects has remained a long-standing problem, with major progress being achieved only recently in joint works with Yair Shenfeld. In this talk I will aim to discuss the history and recent progress on these questions, as well as some of their connections with other branches of mathematics including geometry, analysis, and combinatorics.