Center of maximum-sum matchings of bichromatic points

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