标题： 紧李群中Brunn-Minkowski不等式的非对称稳定性 显示英文标题

标题： Asymmetric stability of the Brunn--Minkowski inequality in compact Lie groups

摘要： 我们证明了最近建立的紧致单李群中的Brunn–Minkowski不等式的稳定性结果。 也就是说，我们证明了：如果紧致单李群$G$中的两个紧致子集$A, B$满足 \$\$ $$ \mu(AB)^{1/d'} \leq (1 + \epsilon)\left(\mu(A)^{1/d'} + \mu(B)^{1/d'}\right)$$ \$\$ 其中$AB$是闵可夫斯基积$\{ab : a \in A, b \in B\}$，$d'$表示适当闭子群的最小余维数，而$\mu$是哈尔测度，那么$A$和$B$必须近似类似于余维数为$d'$的适当子群$H$的邻域，误差取决于$d', \epsilon$和比值$\frac{\mu(A)}{\mu(B)}$。 这一结果表明紧致单李群中的Brunn–Minkowski不等式具有改进的误差率， $$\mu(AB)^{\frac{1}{d'}} \geq (1-C\mu(A)^{\frac{2}{d'}})\left(\mu(A)^{\frac{1}{d'}} + \mu(B)^{\frac{1}{d'}}\right) $$ 尖锐，仅依赖于常数 $C$ ，该常数仅由 $d'$ 和 $\frac{\mu(A)}{\mu(B)}$ 决定。 我们的方法基于作者早期一篇证明Brunn–Minkowski不等式及其在情况 $A=B$ 下稳定性的工作。 我们采用组合多尺度分析，并研究所谓的密度函数。 此外， $A$ 和 $B$ 之间的不对称性引入了新的挑战，需要使用非交换Fourier理论以及Prékopa–Leindler不等式的稳定性结果。 显示英文摘要

摘要： 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 $$ \mu(AB)^{1/d'} \leq (1 + \epsilon)\left(\mu(A)^{1/d'} + \mu(B)^{1/d'}\right)$$ where $AB$ is the Minkowski product $\{ab : a \in A, b \in B\}$, $d'$ denotes the minimal codimension of a proper closed subgroup and $\mu$ 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 $d', \epsilon$ and the ratio $\frac{\mu(A)}{\mu(B)}$. This result implies an improved error rate in the Brunn--Minkowski inequality in compact simple Lie groups $$\mu(AB)^{\frac{1}{d'}} \geq (1-C\mu(A)^{\frac{2}{d'}})\left(\mu(A)^{\frac{1}{d'}} + \mu(B)^{\frac{1}{d'}}\right) $$ sharp, up to the constant $C$ which depends on $d'$ and $\frac{\mu(A)}{\mu(B)}$ 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 Pr\'ekopa--Leindler inequality.