Mathematical Conversations

The Alexandrov-Fenchel inequality---the fundamental log-concavity phenomenon in convex geometry---arose from Minkowski's work in number theory in the late 1800s. It has resurfaced in surprising ways throughout the 20th and 21st centuries in the development of various branches of mathematics, including in spectral theory, algebraic geometry, and combinatorics. I will try to tell some of this story as I understand it, and perhaps some parts I do not understand so well. If time permits, I will also highlight a fundamental question, dating back to the original work of Minkowski, that we resolved some years ago with Yair Shenfeld.