No abstract

For a graph whose vertices are points in R 𝑑 , consider the closed balls with diameters induced by its edges. The graph is called a Tverberg graph if these closed balls intersect.A max-sum tree of a finite point set 𝑋 ⊂ R 𝑑 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 ⊂ R 𝑑 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 R 𝑑 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.Theorem 1. A max-sum tree of any finite point set in R 𝑑 is a Tverberg graph.