Near Equality in the Brunn-Minkowski Inequality

Abstract

A pair of subsets of Euclidean space which nearly achieves equality in the Brunn-Minkowski inequality must nearly coincide with a pair of homothetic convex sets. The two-dimensional case was treated in a previous paper in this series by an argument which does not seem to generalize to higher dimensions. Here the result is extended to arbitrary dimensions. An induction on the dimension, a symmetrization argument, and a description of near solutions of an additive functional equation are used to establish sufficient regularity to set up a compactness argument.

References in corpus (4)

• An Approximate Inverse Riesz-Sobolev Inequality
• Near-extremizers of Young's Inequality for R^d
• Near equality in the two-dimensional Brunn-Minkowski inequality
• Near-extremizers of Young's inequality for discrete groups

Cited by in corpus (8)

• Forward and Reverse Entropy Power Inequalities in Convex Geometry
• Near equality in the two-dimensional Brunn-Minkowski inequality
• Robustness of the Gaussian concentration inequality and the Brunn-Minkowski inequality
• Minimal and nearly minimal measure expansions in connected unimodular groups
• A nonabelian Brunn-Minkowski inequality
• Near equality in the Riesz-Sobolev inequality in higher dimensions
• Quantitative Borell-Brascamp-Lieb inequalities for compactly supported power concave functions (and some applications)
• Quantitative stability for sumsets in $R^n$