On Maximum-Sum Matchings of Bichromatic Points

Huemer et al. (Discrete Math, 2019) proved that for any two finite point sets R and B in the plane with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|R| = |B|$$\end{document}, 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 some family of continuous (semi-)metrics, 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..