The Annals of Probability

Optimal transport from Lebesgue to Poisson

Martin Huesmann and Karl-Theodor Sturm

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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.

Article information

Ann. Probab., Volume 41, Number 4 (2013), 2426-2478.

First available in Project Euclid: 3 July 2013

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 49Q20: Variational problems in a geometric measure-theoretic setting

Optimal transportation fair allocation Laguerre tessellation Poisson point process


Huesmann, Martin; Sturm, Karl-Theodor. Optimal transport from Lebesgue to Poisson. Ann. Probab. 41 (2013), no. 4, 2426--2478. doi:10.1214/12-AOP814.

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