Asymmetric stability of the Brunn--Minkowski inequality in compact Lie groups
Simon Machado
TL;DR
The paper proves a quantitative asymmetric stability result for the Brunn–Minkowski inequality in compact simple Lie groups: if two compact sets $A,B\subset G$ satisfy $\mu(AB)^{1/d'}\le (1+\varepsilon)(\mu(A)^{1/d'}+\mu(B)^{1/d'})$ with $d'$ the minimal codimension of a proper closed subgroup, then $A$ and $B$ are close to neighborhoods of a closed subgroup $H$ of codimension $d'$, with explicit dependence on $d',\varepsilon$ and the ratio $\mu(A)/\mu(B)$. The approach blends local stability in small balls (via density functions, multi-scale analysis, and the stability of the Prékopa–Leindler inequality) with a global reduction that leverages non-abelian Fourier analysis and a combinatorial lemma to pass from local structure near a subgroup to a global extremal description. The results yield an improved error rate in the global Brunn–Minkowski inequality, showing that near-extremizers must concentrate near subgroups, and provide a concrete quantitative bridge between Euclidean convexity intuition and the non-abelian group setting. The work advances the understanding of near-extremal configurations in additive combinatorics on compact Lie groups and supplies tools that may impact related stability results and structure theorems on groups.
Abstract
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 $$ μ(AB)^{1/d'} \leq (1 + ε)\left(μ(A)^{1/d'} + μ(B)^{1/d'}\right)$$ where $AB$ is the Minkowski product $\{ab : a \in A, b \in B\}$, $d'$ 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 $\frac{μ(A)}{μ(B)}$. This result implies an improved error rate in the Brunn--Minkowski inequality in compact simple Lie groups $$μ(AB)^{\frac{1}{d'}} \geq (1-Cμ(A)^{\frac{2}{d'}})\left(μ(A)^{\frac{1}{d'}} + μ(B)^{\frac{1}{d'}}\right) $$ sharp, up to the constant $C$ which depends on $d'$ and $\frac{μ(A)}{μ(B)}$ 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.