Sharp stability of the Brunn-Minkowski inequality via optimal mass transportation
| Autor: | Figalli, Alessio, van Hintum, Peter, Tiba, Marius |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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). Comment: 25 pages |
| Databáze: | arXiv |
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