Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, 2005. MR 2129498 (2006k:49001)
  • 2. R. M. Dudley, Real Analysis and Probability, Cambridge Studies in Advanced Mathematics 74, Cambridge University Press, 2002. MR 1932358 (2003h:60001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 60B05

Retrieve articles in all journals with MSC (2000): 60B05

Additional Information

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

Wolfgang Desch
Affiliation: Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstrasse 36, 8010 Graz, Austria

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

American Mathematical Society