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• W3023062724 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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C upper D left-parenthesis upper K comma upper N right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>CD(K,N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -space or an essentially non-branching <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M upper C upper P left-parenthesis upper K comma upper N right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>MCP(K,N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M upper C upper P left-parenthesis upper K comma upper N right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>MCP(K,N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-spaces." @default.
• W3023062724 created "2020-05-13" @default.
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• W3023062724 date "2020-10-21" @default.
• W3023062724 modified "2023-09-27" @default.
• W3023062724 title "On one-dimensionality of metric measure spaces" @default.
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• W3023062724 doi "https://doi.org/10.1090/proc/15162" @default.
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