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$\mathbb R^d$中的点的图，考虑由其边作为直径诱导的闭球。若这些闭球相交，则该图称为Tverberg图。有限点集$X \subset \mathbb R^d$的最大和树是以$X$为顶点集的树，在所有以$X$为顶点集的树中，其边欧几里得距离之和最大。类似地，偶数点集$X \subset \mathbb R^d$的最大和匹配是$X$的完美匹配，在所有$X$的完美匹配中，匹配点对之间的欧几里得距离之和最大。我们证明了$\mathbb R^d$中任意有限点集的最大和树是Tverberg图，这推广了Abu-Affash等人最近在平面上建立该结论的结果。此外，我们给出了Bereg等人定理的一个新证明，该定理指出平面上任意偶数点集的最大和匹配是Tverberg图。并且，我们证明了该定理的一个略强版本。