On the Existence of Monge Solutions to Multi-marginal Optimal Transport with Quadratic Cost and Uniform Discrete Marginals

A natural and important question in multi-marginal optimal transport is whether the Monge ansatz is justified; does there exist a solution of Monge, or deterministic, form? We address this question for the quadratic cost when each marginal measure is m-empirical (that is, uniformly supported on m points). By direct computation, we provide an example showing that the ansatz can fail when the underlying dimension d is 2, the number of marginals N to be matched is 3, and the size m of their supports is 3. As a consequence, the set of m-empirical measures is not barycentrically convex when N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 3$$\end{document}, d≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \ge 2$$\end{document}, and m≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 3$$\end{document}. It is a well-known consequence of the Birkhoff-von Neumann theorem that the Monge ansatz holds for N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2$$\end{document}, standard techniques show it holds when d=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1$$\end{document}, and we provide a simple proof here that it holds whenever m=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=2$$\end{document}. Therefore, the N, d, and m in our counterexample are as small as possible.