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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43069_2025_437_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$N \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43069_2025_437_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$d \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43069_2025_437_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$m \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. It is a well-known consequence of the Birkhoff-von Neumann theorem that the Monge ansatz holds for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43069_2025_437_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$N=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, standard techniques show it holds when <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43069_2025_437_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$d=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and we provide a simple proof here that it holds whenever <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43069_2025_437_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$m=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Therefore, the N, d, and m in our counterexample are as small as possible.

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

期刊论文
发表日期：

2025年4月7日

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Pedram Emami,
Brendan Pass

摘要

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 $N \ge 3$ $N \geq 3$ , $d \ge 2$ $d \geq 2$ , and $m \ge 3$ $m \geq 3$ . It is a well-known consequence of the Birkhoff-von Neumann theorem that the Monge ansatz holds for $N=2$ $N = 2$ , standard techniques show it holds when $d=1$ $d = 1$ , and we provide a simple proof here that it holds whenever $m=2$ $m = 2$ . Therefore, the N, d, and m in our counterexample are as small as possible.