An elementary proof of the triangle inequality for the Wasserstein metric
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- by Philippe Clement and Wolfgang Desch PDF
- Proc. Amer. Math. Soc. 136 (2008), 333-339 Request permission
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
- 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
- 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, DOI 10.1017/CBO9780511755347
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
- Received by editor(s): October 30, 2006
- Published electronically: September 27, 2007
- Communicated by: Richard C. Bradley
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 333-339
- MSC (2000): Primary 60B05
- DOI: https://doi.org/10.1090/S0002-9939-07-09020-X
- MathSciNet review: 2350420