Sharp quantitative stability of the Brunn-Minkowski inequality

2023/10/31 by Alessio Figalli, Figalli, Alessio, Peter van Hintum +3 · 3 citations
Computer Science · Mathematics · #49Q20 #52A27 #52A40 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Equations Stability Results #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2310.20643

openalex publication_date 2023/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Brunn-Minkowski inequality states that for bounded measurable sets A and B in ℝⁿ, we have |A+B|^1/n ≥ |A|^1/n+|B|^1/n. Also, equality holds if and only if A and B are convex and homothetic sets in ℝ^d. The stability of this statement is a well-known problem that has attracted much attention in recent years. This paper gives a conclusive answer by proving the sharp stability result for the Brunn-Minkowski inequality on arbitrary sets.

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