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Nonpositive curvature and the Ptolemy inequality


Foertsch, T; Lytchak, A; Schroeder, V (2007). Nonpositive curvature and the Ptolemy inequality. International Mathematics Research Notices, 2007(rnm100):online.

Abstract

We provide examples of nonlocally, compact, geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally, compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolemy.

We provide examples of nonlocally, compact, geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally, compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolemy.

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9 citations in Web of Science®
15 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2007
Deposited On:10 Dec 2009 15:55
Last Modified:05 Apr 2016 13:23
Publisher:Oxford University Press
ISSN:1073-7928
Additional Information:This is a pre-copy-editing, author-produced PDF of an article accepted for publication in International Mathematics Research Notices following peer review. The definitive publisher-authenticated version "Foertsch, T; Lytchak, A; Schroeder, V (2007). Nonpositive curvature and the Ptolemy inequality. International Mathematics Research Notices, 2007(rnm100)" is available online at: International Mathematics Research Notices
Publisher DOI:https://doi.org/10.1093/imrn/rnm100
Related URLs:http://arxiv.org/abs/0705.1042
Permanent URL: https://doi.org/10.5167/uzh-21536

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