Abstract:In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corollary we obtain that if a metric measure space is a very strict -space or an essentially non-branching -space with some open set isometric to an interval, then it is a one-dimensional manifold. We also obtain the same conclusion for a metric measure space which has a point in which the Gromov-Hausdorff tangent is unique and isometric to the real line, and for which the optimal transport maps not only exist but are unique. Again, we obtain an analogous corollary in the setting of essentially non-branching -spaces.

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中文翻译：

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关于度量尺度空间的一维性

摘要：本文证明了一个度量测度空间是至少一个等距到一个区间的开放集，并且从任何绝对连续测度到任意测度都存在（可能是非唯一的）最优输运图一维流形（可能带有边界）。作为直接的推论，我们得到的是，如果度量度量空间是非常严格的空间或本质上是非分支的-具有一定等距间隔的开放集的空间，那么它是一维流形。对于度量度量空间，我们也得出相同的结论，该度量度量空间的Gromov-Hausdorff切线是唯一的，并且与实线等距，并且对于最优运输图，不仅存在而且是唯一的。同样，我们在本质上非分支空间的设置中获得了类似的推论。

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