Center of maximum-sum matchings of bichromatic points

Abstract: Let $R$ and $B$ be two disjoint point sets in the plane with $|R|=|B|=n$. Let $\mathcal{M}={(r_i,b_i),i=1,2,\ldots,n}$ be a perfect matching that matches points of $R$ with points of $B$ and maximizes $\sum_{i=1}<sup>n|r_i-b_i|$, 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 $\mathcal{M}$) such that $|r_i-o|+|b_i-o|\le \sqrt{2}~|r_i-b_i|$ for all $i\in{1,2,\ldots,n}$.