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On one-dimensionality of metric measure spaces

Year of publication

2021

Authors

Schultz, Timo

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 CD(K, N) -space or an essentially non-branching MCP(K, N)-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 MCP(K, N)-spaces
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Organizations and authors

Publication type

Publication format

Article

Parent publication type

Journal

Article type

Original article

Audience

Scientific

Peer-reviewed

Peer-Reviewed

MINEDU's publication type classification code

A1 Journal article (refereed), original research

Publication channel information

Volume

149

Issue

1

Pages

383-396

​Publication forum

65443

​Publication forum level

2

Open access

Open access in the publisher’s service

No

Self-archived

Yes

Other information

Fields of science

Mathematics

Keywords

[object Object],[object Object],[object Object]

Publication country

United States

Internationality of the publisher

International

Language

English

International co-publication

No

Co-publication with a company

No

DOI

10.1090/proc/15162

The publication is included in the Ministry of Education and Culture’s Publication data collection

Yes