Intersecting balls induced by a geometric graph II
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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.
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