2024/11/15 by Károly Bezdek, Bezdek, Károly, Zsolt Lángi +3 · 3 citations
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #FOS: Mathematics #Mathematics and Applications #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2411.10302
openalex publication_date 2024/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this survey, we discuss volumetric and combinatorial results concerning (mostly finite) intersections or unions of balls (mostly of equal radii) in the d-dimensional real vector space, mostly equipped with the Euclidean norm. Our first topic is the Kneser--Poulsen Conjecture, according to which if a finite number of balls are rearranged so that the pairwise distances of the centers increase, then the volume of the union (resp., intersection) increases (resp., decreases). Next, we discuss Blaschke--Santaló-type inequalities, and reverse isoperimetric inequalities for convex sets in Euclidean d-space obtained as intersections of (possibly infinitely many) balls of radius r, which we call r-ball bodies. We present some results on 1-ball bodies (also called ball-bodies or spindle convex sets) in the plane, with special attention paid to their approximation by the spindle convex hull of a finite subset. A ball-polyhedron is a ball-body obtained as the intersection of finitely many unit balls in Euclidean d-space. We consider the combinatorial structure of their faces, and volumetric properties of ball-polyhedra obtained from choosing the centers of the balls randomly.