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
Show moreOrganizations and authors
University of Jyväskylä
Schultz Timo
Publication type
Publication format
Article
Parent publication type
Journal
Article type
Original article
Audience
ScientificPeer-reviewed
Peer-ReviewedMINEDU's publication type classification code
A1 Journal article (refereed), original researchPublication channel information
Publisher
Volume
149
Issue
1
Pages
383-396
ISSN
Publication forum
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