For a graph whose vertices are points in $\mathbb R^d$, 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 $X \subset \mathbb R^d$ 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 $X \subset \mathbb R^d$ 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 $\mathbb R^d$ 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.

翻译：对于顶点位于 $\mathbb R^d$ 中的图，考虑以其边为直径的闭球。若这些闭球相交，则该图称为 Tverberg 图。对于有限点集 $X \subset \mathbb R^d$，最大和树是以 $X$ 为顶点集，在所有以 $X$ 为顶点集的树中最大化边欧氏距离之和的树。类似地，对于偶数点集 $X \subset \mathbb R^d$，最大和匹配是 $X$ 的完美匹配中最大化匹配点间欧氏距离之和的完美匹配。我们证明，$\mathbb R^d$ 中任意有限点集的最大和树均为 Tverberg 图，这推广了 Abu-Affash 等人近期在平面情形下的结论。此外，我们给出了 Bereg 等人定理的一个新证明，该定理指出平面中任意偶数点集的最大和匹配是 Tverberg 图。同时，我们证明了该定理的一个稍强版本。