2025/04/01 by Simon Machado, Machado, Simon
Mathematics · #Point processes and geometric inequalities #Geometry and complex manifolds #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2504.00895
We show a stability result for the recently established Brunn--Minkowski inequality in compact simple Lie groups. Namely, we prove that if two compact subsets A, B of a compact simple Lie group G satisfy μ(AB)1/d' ≤ (1 + ε)(μ(A)1/d' + μ(B)1/d') where AB is the Minkowski product \ab : a ∈ A, b ∈ B\, d' denotes the minimal codimension of a proper closed subgroup and μ is a Haar measure, then A and B must approximately look like neighbourhoods of a proper subgroup H of codimension d', with an error that depends quantitatively on d', ε and the ratio (μ(A))/(μ(B)). This result implies an improved error rate in the Brunn--Minkowski inequality in compact simple Lie groups μ(AB)(1)/(d') ≥ (1-Cμ(A)(2)/(d'))(μ(A)(1)/(d') + μ(B)(1)/(d')) sharp, up to the constant C which depends on d' and (μ(A))/(μ(B)) alone. Our approach builds upon an earlier paper of the author proving the Brunn--Minkowski inequality, and stability in the case A=B. We employ a combinatorial multi-scale analysis and study so-called density functions. Additionally, the asymmetry between A and B introduces new challenges, requiring the use of non-abelian Fourier theory and stability results for the Prékopa--Leindler inequality.