Near Equality in the Brunn-Minkowski Inequality

TL;DR

This paper extends the understanding of near equality cases in the Brunn-Minkowski inequality from two dimensions to all higher dimensions, using induction, symmetrization, and functional equations.

Contribution

It generalizes previous two-dimensional results to arbitrary dimensions through new techniques and a compactness argument.

Findings

Near equality implies sets are close to homothetic convex sets

The method applies induction and symmetrization in higher dimensions

Provides a characterization of near solutions to an additive functional equation

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.