2012/06/09 by Michael Christ, Christ, Michael · 3 citations
Mathematics · #52A40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Inequalities and Applications #Point processes and geometric inequalities #math.CA #msc:52A40
paper · pdf · doi:10.48550/arxiv.1206.1965
openalex publication_date 2012/06/09 · arxiv created 2012/07/20 · arxiv updated 2012/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If a pair of subsets of two-dimensional Euclidean space nearly achieves equality in the Brunn-Minkowski inequality, in the sense that the measure of the associated sumset is nearly equal to the lower bound provided by the inequality, then these sets must nearly coincide with a pair of homothetic convex sets. The proof relies on a continuum analogue of a theorem of Freiman which characterizes finite sets of integers whose sumsets are of nearly minimal size. Small corrections and clarifications have been made in this draft.