Intersecting diametral balls induced by a geometric graph II

Abstract

For a graph whose vertices are points in Rd, consider the closed balls with diameters induced by its edges. The graph is called a Tverberg graph if these closed balls share a common point.A max-sum tree of a finite point set X⊂Rd is a tree with vertex set X that maximizes the sum of Euclidean distances of its edges among all trees with vertex set X. Similarly, a max-sum matching of an even set X⊂Rd is a perfect matching of X maximizing the sum of Euclidean distances between the matched points among all perfect matchings of X.We prove that a max-sum tree of any finite point set in Rd is a Tverberg graph, which generalizes a recent result of Abu-Affash et al., who established this claim in the plane. Additionally, we provide a new proof of a theorem by Bereg et al., which states that a max-sum matching of any even point set in the plane is a Tverberg graph. Moreover, we proved a slightly stronger version of this theorem.