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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An elementary proof of the triangle inequality for the Wasserstein metric


Authors: Philippe Clement and Wolfgang Desch
Journal: Proc. Amer. Math. Soc. 136 (2008), 333-339
MSC (2000): Primary 60B05
Published electronically: September 27, 2007
MathSciNet review: 2350420
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Abstract: We give an elementary proof for the triangle inequality of the $ p$-Wasserstein metric for probability measures on separable metric spaces. Unlike known approaches, our proof does not rely on the disintegration theorem in its full generality; therefore the additional assumption that the underlying space is Radon can be omitted. We also supply a proof, not depending on disintegration, that the Wasserstein metric is complete on Polish spaces.


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Additional Information

Philippe Clement
Affiliation: Mathematical Institute, Leiden University, P. O. Box 9512, NL-2300 RA Leiden, The Netherlands
Email: philippeclem@gmail.com

Wolfgang Desch
Affiliation: Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstrasse 36, 8010 Graz, Austria
Email: georg.desch@uni-graz.at

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09020-X
PII: S 0002-9939(07)09020-X
Keywords: Wasserstein metric, triangle inequality, probability measures on metric spaces
Received by editor(s): October 30, 2006
Published electronically: September 27, 2007
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2007 American Mathematical Society