Sharp stability of the Brunn-Minkowski inequality via optimal mass transportation

2024/07/15 by Alessio Figalli, Figalli, Alessio, Peter van Hintum +3 · 3 citations
Mathematics · #Point processes and geometric inequalities #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2407.10932

Abstract

The Brunn-Minkowski inequality, applicable to bounded measurable sets A and B in ℝ^d, states that |A+B|^1/d ≥ |A|^1/d+|B|^1/d. Equality is achieved if and only if A and B are convex and homothetic sets in 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).

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