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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 | 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. Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. MR 2129498
  • 2. R. M. Dudley, Real analysis and probability, Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002. Revised reprint of the 1989 original. MR 1932358

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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