Matching points with disks with a common intersection

Abstract

We consider matchings with diametral disks between two sets of points R and B. More precisely, for each pair of matched points p ∈ R and q ∈ B, we consider the disk through p and q with the smallest diameter. We prove that for any R and B such that |R| = |B|, there exists a perfect matching such that the diametral disks of the matched point pairs have a common intersection. In fact, our result is stronger, and shows that a maximum weight perfect matching has this property.

On Maximum-Sum Matchings of Points

“…Our results, as that of [14], are in the direction opposite to that of above mentioned known results on matching points: The goal is that all matching objects (i.e., the disks in D M ) have a common intersection, whereas in previous work it is required that all matching objects are pairwise disjoint.…”

“…As described by Huemer et al [14], this class of problems is well studied in discrete and computational geometry, starting from the classic result that n red points and n blue points can always be perfectly matched with n pairwise non-crossing segments, where each segment connects a red point with a blue point [16]. The study has been continued in plenty of directions, for both the monochromatic and bichromatic versions, by using pairwise disjoint objects inducing the matching: segments [4,11], rectangles [1,2,6,7,9], and more general geometric objects [5].…”

“…Huemer et al [14] proved that if M is any matching that maximizes the total squared Euclidean distance of the matched points, i.e., it maximizes (p,q)∈M p − q 2 , then all disks of D M have a point in common.…”

Intersecting diametral balls induced by a geometric graph

“…It claims that for any set of (r − 1)(d + 1) + 1 points in R d , there exists a partition of points into r parts whose convex hulls intersect. In the current paper, we consider a variation of Tverberg's problem introduced recently in [2,5,8].…”

“…Let G be a graph whose vertex set is a finite set of points in R d . We say that G is a Tverberg graph if In 2019, Huemer, Pérez-Lantero, Seara, and Silveira [5] showed that for any n red points and any n blue points in the plane, there is a red-blue Tverberg matching (every edge of this Tverberg matching connects a red vertex with a blue one). Later, Bereg, Chacón-Rivera, Flores-Peñaloza, Huemer, and Pérez-Lantero [2] found a second proof of the monochromatic version of this result, that is, for any 2n points in the plane, there is a Tverberg matching.…”