Center of maximum-sum matchings of bichromatic points

Abstract

Let R and B be two point sets in the plane with |R|=|B|=n. Let M={(ri,bi),i=1,2,…,n} be a perfect matching that matches points of R with points of B and maximizes ∑i=1n‖ri−bi‖, the total Euclidean distance of the matched pairs. In this paper, we prove that there exists a point o of the plane (the center of M) such that ‖ri−o‖+‖bi−o‖≤2‖ri−bi‖ for all i∈{1,2,…,n}.