Center of maximum-sum matchings of bichromatic points

Let $R$ and $B$ be two disjoint point sets in the plane with $\mathrm{\mid }R\mathrm{\mid }=\mathrm{\mid }B\mathrm{\mid }=n$. Let $\mathcal{M}=\{(r_i,b_i),i=1,2,\dots ,n\}$ be a perfect matching that matches points of $R$ with points of $B$ and maximizes ${\sum }_{i=1}^n\mathrm{\parallel }{r}_i-b_i\mathrm{\parallel }$, 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 $\mathrm{\parallel }{r}_i-o\mathrm{\parallel }+\mathrm{\parallel }{b}_i-o\mathrm{\parallel }\le \sqrt{2}\mathrm{\parallel }{r}_i-b_i\mathrm{\parallel }$ for all $i\in \{1,2,\dots ,n\}$.