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July 2013 Optimal transport from Lebesgue to Poisson
Martin Huesmann, Karl-Theodor Sturm
Ann. Probab. 41(4): 2426-2478 (July 2013). DOI: 10.1214/12-AOP814


This paper is devoted to the study of couplings of the Lebesgue measure and the Poisson point process. We prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite. Moreover, we give precise conditions for the latter which demonstrate a sharp threshold at $d=2$. The cost will be defined in terms of an arbitrary increasing function of the distance.

The coupling will be realized by means of a transport map (“allocation map”) which assigns to each Poisson point a set (“cell”) of Lebesgue measure 1. In the case of quadratic costs, all these cells will be convex polytopes.


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Martin Huesmann. Karl-Theodor Sturm. "Optimal transport from Lebesgue to Poisson." Ann. Probab. 41 (4) 2426 - 2478, July 2013.


Published: July 2013
First available in Project Euclid: 3 July 2013

zbMATH: 1279.60024
MathSciNet: MR3112922
Digital Object Identifier: 10.1214/12-AOP814

Primary: 60D05
Secondary: 49Q20 , 52A22

Keywords: fair allocation , Laguerre tessellation , Optimal transportation , Poisson point process

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 4 • July 2013
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