2010/12/31 by Martin Huesmann, Karl-Theodor Sturm · 1 citation
Mathematics · #Coupling (piping) #Geometric Analysis and Curvature Flows #Lebesgue integration #Lebesgue measure #Measure (data warehouse) #Point processes and geometric inequalities #Poisson distribution #Quadratic equation #Random Matrices and Applications #Regular polygon #Uniqueness #math.PR
paper · pdf · doi:10.1214/12-aop814
published as Annals of Probability 2013, Vol. 41, No. 4, 2426-2478 · Published in at http://dx.doi.org/10.1214/12-AOP814 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2013/07/01 · arxiv created 2013/08/13 · arxiv updated 2013/08/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.