Simon Machado ETH Zurich

Brunn–Minkowski inequality in Compact Lie Groups

Abstract:

The Brunn–Minkowski inequality is a cornerstone of convex geometry and analysis, describing how volume behaves under Minkowski addition. But what becomes of this principle in non-abelian settings—such as SO_n(ℝ)—where addition is replaced by group multiplication? In this talk, I will present a natural, yet only recently established, analogue of the Brunn--Minkowski inequality on compact Lie groups, answering a question posed by Breuillard and Green. Specifically, in SO_n(ℝ), the inequality takes the form

μ(AB)^1/(n-1) ≥ (1 - ε)(μ(A)^1/(n-1) + μ(B)^1/(n-1)),

where μ denotes Haar measure, AB = {ab : a ∈ A, b ∈ B}, and ε is a small corrective factor that reflects the effects of non-negative curvature. I will focus in particular on stability properties of this inequality. While the absence of commutativity and linear structure presents significant challenges, techniques from the study of product growth in groups allow us to reduce the problem to a form accessible to analytic tools—such as the Prékopa–Leindler inequality, optimal transport, spherical harmonics, and multi-scale analysis. This perspective sheds new light on geometric and probabilistic properties of compact groups—connecting to questions about isoperimetry, diameter and growth estimates, and mixing times of random walks—and suggests a broader framework of geometric inequalities in non-abelian settings.