Sharp stability of the Brunn-Minkowski inequality via optimal mass transportation
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909255649460224 |
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| author | Figalli, Alessio van Hintum, Peter Tiba, Marius |
| author_facet | Figalli, Alessio van Hintum, Peter Tiba, Marius |
| contents | The Brunn-Minkowski inequality, applicable to bounded measurable sets $A$ and $B$ in $\mathbb{R}^d$, states that $|A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}$. Equality is achieved if and only if $A$ and $B$ are convex and homothetic sets in $\mathbb{R}^d$. The concept of stability in this context concerns how, when approaching equality, sets $A$ and $B$ are close to homothetic convex sets. In a recent breakthrough [FvHT23], the authors of this paper proved the following folklore conjectures on the sharp stability for the Brunn-Minkowski inequality:
(1) A linear stability result concerning the distance from $A$ and $B$ to their respective convex hulls.
(2) A quadratic stability result concerning the distance from $A$ and $B$ to their common convex hull.
As announced in [FvHT23], in the present paper, we leverage (1) in conjunction with a novel optimal transportation approach to offer an alternative proof for (2). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10932 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp stability of the Brunn-Minkowski inequality via optimal mass transportation Figalli, Alessio van Hintum, Peter Tiba, Marius Analysis of PDEs Combinatorics Metric Geometry 52A40, 49Q22, 49Q20, 52A27 The Brunn-Minkowski inequality, applicable to bounded measurable sets $A$ and $B$ in $\mathbb{R}^d$, states that $|A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}$. Equality is achieved if and only if $A$ and $B$ are convex and homothetic sets in $\mathbb{R}^d$. The concept of stability in this context concerns how, when approaching equality, sets $A$ and $B$ are close to homothetic convex sets. In a recent breakthrough [FvHT23], the authors of this paper proved the following folklore conjectures on the sharp stability for the Brunn-Minkowski inequality: (1) A linear stability result concerning the distance from $A$ and $B$ to their respective convex hulls. (2) A quadratic stability result concerning the distance from $A$ and $B$ to their common convex hull. As announced in [FvHT23], in the present paper, we leverage (1) in conjunction with a novel optimal transportation approach to offer an alternative proof for (2). |
| title | Sharp stability of the Brunn-Minkowski inequality via optimal mass transportation |
| topic | Analysis of PDEs Combinatorics Metric Geometry 52A40, 49Q22, 49Q20, 52A27 |
| url | https://arxiv.org/abs/2407.10932 |