Quantitative stability for the Brunn-Minkowski inequality

By Alessio Figalli, David Jerison. 2014

94 citations recorded by the source corpus.

We prove a quantitative stability result for the Brunn-Minkowski inequality: if $|A|=|B|=1$, $t \in [\tau,1-\tau]$ with $\tau>0$, and $|tA+(1-t)B|^{1/n}\leq 1+\delta$ for some small $\delta$, then, up to a translation, both $A$ and $B$ are quantitatively close (in terms of $\delta$) to a convex set $K$.

Subfields: Applied Mathematics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/6a920eb5da17b1aed39063ec21954622f883808a

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In Applied Mathematics

1. Max Fathi
2. E. Indrei
3. Y. Zhang
4. David Jerison
5. Michel Ledoux
6. Alessio Figalli
7. Luigi Ambrosio
8. Guido De Philippis

In Molecular Biology

1. David Jerison
2. E. Indrei
3. Alessio Figalli
4. Michel Ledoux
5. Max Fathi
6. Y. Zhang

Citations in the corpus

• Quantitative logarithmic Sobolev inequalities and stability estimates (2014, result): “Examples of such results are the recent quantitative stability estimates for Sobolev [12, 19], Brunn-Minkowski [17, 16], and isoperimetric inequalities [20, 18, 15, 23].”
• Sharp gradient stability for the Sobolev inequality (2020, background): “Introduction Motivated by important applications to problems in the calculus of variations and evolution PDEs, in recent years there has been a growing interest around the understanding of quantitative stability for functional/geometric inequalities, see for instance [3, 2, 8, 27, 28, 21, 9, 22, 29,”
• Quantitative stability for sumsets in $R^n$ (2014, background): “For use in the sequel [5], we observe that if δ(A) ≤ δn, then there exists a convex set K such that δ(A)n ≥ cn |K∆A| |A| , αn := 1 8 · 16n−2n!(n− 1)! .”