The classical Brunn--Minkowski inequality is a cornerstone of convex geometry and analysis, relating the volume of Minkowski sums via

$$\lambda(A+B)^{1/d} \geq \lambda(A)^{1/d} + \lambda(B)^{1/d}.$$ 

I will present an analogue of this inequality for compact Lie groups (e.g., $SO_n(\mathbb{R})$), resolving a conjecture of Breuillard and Green. In this setting, vector addition is replaced by group multiplication, and new phenomena linked to positive curvature arise.

The approach combines tools from group combinatorics and Euclidean analysis—particularly optimal transport and the stability theory of the Prékopa--Leindler inequality—with methods involving spherical harmonics and multi-scale analysis à la Bourgain.

I will also discuss links to open problems on the geometry of compact groups and their homogeneous spaces, including isoperimetry, diameter and growth estimates.

Café culturel à 13h05 servi par Bruno Duchesne.