Optimal transport from Lebesgue to Poisson

TL;DR

This paper investigates the optimal way to couple Lebesgue measure with Poisson point processes, establishing existence, uniqueness, and geometric properties of the transport map under various cost functions, with a focus on quadratic costs.

Contribution

It proves the existence and uniqueness of optimal couplings between Lebesgue measure and Poisson processes, characterizes the structure of the transport map, and identifies a sharp dimension threshold for cost finiteness.

Findings

Existence and uniqueness of optimal couplings when cost is finite.

Transport cells are convex polytopes for quadratic costs.

A sharp threshold at dimension d=2 for cost finiteness.

Abstract

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.