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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.00895 |
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| _version_ | 1866909560741036032 |
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| author | Machado, Simon |
| author_facet | Machado, Simon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00895 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymmetric stability of the Brunn--Minkowski inequality in compact Lie groups Machado, Simon Group Theory 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. |
| title | Asymmetric stability of the Brunn--Minkowski inequality in compact Lie groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2504.00895 |