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Limit theorems for optimal mass transportation

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Abstract

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 λ+, λ− of equal mass

$$W_1(\lambda^-, \lambda^+)= \lim_{\varepsilon\rightarrow 0} \varepsilon^{-1} \inf_{\mu} W_p(\mu+\varepsilon\lambda^-, \mu+\varepsilon\lambda^+) $$

where W p , p ≥ 1 is the Wasserstein distance and the infimum is over the set of probability measures in the ambient space.

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  1. Department of Mathematics, Technion, Haifa, 32000, Israel

    G. Wolansky

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  1. G. Wolansky
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Correspondence to G. Wolansky.

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Communicated by L. Ambrosio.

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Wolansky, G. Limit theorems for optimal mass transportation. Calc. Var. 42, 487–516 (2011). https://doi.org/10.1007/s00526-011-0395-x

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