Bi-Lipschitz embeddings of the space of unordered $m$-tuples with a partial transportation metric

TL;DR

This paper constructs bi-Lipschitz embeddings of unordered m-tuples in a space of measures with a partial transportation metric into Hilbert space, extending Almgren's theorem to optimal partial transport.

Contribution

It generalizes Almgren's bi-Lipschitz embedding theorem to the setting of the partial transportation metric on measures.

Findings

Wb(Ω) is isometric to a subset of measures with Wasserstein distance on a completed space.

Constructs bi-Lipschitz embeddings of unordered m-tuples into Hilbert space.

Extends classical embedding results to optimal partial transport setting.

Abstract

Let $\Omega\subset \mathbb{R}^n$ be non-empty, open and proper. Consider $Wb(\Omega)$, the space of finite Borel measures on $\Omega$ equipped with the partial transportation metric introduced by Figalli and Gigli that allows the creation and destruction of mass on $\partial \Omega$. Equivalently, we show that $Wb(\Omega)$ is isometric to a subset of all Borel measures with the ordinary Wasserstein distance, on the one point completion of $\Omega$ equipped with the shortcut metric \[\delta(x,y)= \min\{\|x-y\|, \operatorname{dist}(x,\partial \Omega)+\operatorname{dist}(y,\partial\Omega)\}.\] In this article we construct bi-Lipschitz embeddings of the set of unordered $m$-tuples in $Wb(\Omega)$ into Hilbert space. This generalises Almgren's bi-Lipschitz embedding theorem to the setting of optimal partial transport.