Huemer et al. (Discrete Math, 2019) proved that for any two finite point sets $R$ and $B$ in the plane with $|R| = |B|$, the perfect matching that matches points of $R$ with points of $B$, and maximizes the total squared Euclidean distance of the matched pairs, has the property that all the disks induced by the matching have a nonempty common intersection. A pair of matched points induces the disk that has the segment connecting the points as diameter. In this note, we characterize these maximum-sum matchings for any continuous (semi)metric, focusing on both the Euclidean distance and squared Euclidean distance. Using this characterization, we give a different but simpler proof for the common intersection property proved by Huemer et al..

翻译：Huemer等人（Discrete Math, 2019）证明：对于平面上任意两个有限点集$R$和$B$且满足$|R| = |B|$，将$R$中点与$B$中点点对点匹配，并使匹配对欧氏距离平方和最大的完美匹配，具有以下性质：该匹配诱导的所有圆盘存在非空公共交集。其中，每一对匹配点以连接这两点的线段为直径诱导一个圆盘。本文针对任意连续（半）度量，聚焦于欧氏距离和欧氏距离平方，刻画了这些最大和匹配的特征。利用该刻画方法，我们为Huemer等人证明的公共交集性质提供了一种不同但更简洁的证明。