论文标题
循环成本的多边界最佳运输问题
Multi-marginal optimal transportation problem for cyclic costs
论文作者
论文摘要
我们研究了一个多边界最佳运输问题，具有表格$ c（x_ {1}，\ ldots，x_ {m}）= \ sum_ {k = 1}^{m-1} | x__ {k} -x_ {k} -x_ { f（x_ {1}）|^{2} $，其中$ f：\ mathbb {r}^n \ rightArrow \ mathbb {r}^n $。当$ m = 4 $，$ f $是身份映射的积极倍数，对于Lebesgue度量，第一个也是最后一个边缘是绝对连续的，我们确定Kantorovich问题的任何解决方案都是由地图引起的；因此，解决方案是唯一的。我们继续表明，从某种意义上说，这个结果是敏锐的。确切地说，我们展示的例子表明，如果有以下任何一个，则可以集中于更高的维度集：1）$ f $除了身份的正标量倍数之外，其他任何线性映射，2）最后一个边缘不是绝对连续的，相对于Lebesgie度量，或者是Marginals Marginals $ m \ geq 5 $，甚至是$ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $。
We study a multi-marginal optimal transportation problem with a cost function of the form $c(x_{1}, \ldots,x_{m})=\sum_{k=1}^{m-1}|x_{k}-x_{k+1}|^{2} + |x_{m}- F(x_{1})|^{2}$, where $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$. When $m=4$, $F$ is a positive multiple of the identity mapping, and the first and last marginals are absolutely continuous with respect to Lebesgue measure, we establish that any solution of the Kantorovich problem is induced by a map; the solution is therefore unique. We go on to show that this result is sharp in a certain sense. Precisely, we exhibit examples showing that Kantorovich solutions may concentrate on a higher dimensional sets if any of the following hold: 1) $F$ is any linear mapping other than a positive scalar multiple of the identity, 2) the last marginal is not absolutely continuous with respect to Lebesgue measure, or 3) the number of marginals $m \geq 5$, even when $F$ is the identity mapping.
