Gershon Wolansky
Published 2009-03-01, updated 2010-10-18Version 5
The optimal mass transportation was introduced by Monge some 200 years ago and is, today, the source of large number of results in analysis, geometry and convexity. Here I investigate a new, surprising link between optimal transformations obtained by different Lagrangian actions on Riemannian manifolds. As a special case, for any pair of non-negative measures $\lambda^+,\lambda^-$ of equal mass $$ W_1(\lambda^-, \lambda^+)= \lim_{\eps\to 0} \eps^{-1}\inf_{\mu} W_p(\mu+\eps\lambda^-, \mu+\eps\lambda^+)$$ where $W_p$, $p\geq 1$ is the Wasserstein distance and the infimum is over the set of probability measures in the ambient space.
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