Near Equality in the Brunn-Minkowski Inequality
Classical Analysis and ODEs
2012-07-24 v1
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.
Keywords
Cite
@article{arxiv.1207.5062,
title = {Near Equality in the Brunn-Minkowski Inequality},
author = {Michael Christ},
journal= {arXiv preprint arXiv:1207.5062},
year = {2012}
}