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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\emph{partial} transportation metric introduced by Figalli and Gigli thatallows the creation and destruction of mass on $\partial \Omega$. Equivalently,we show that $Wb(\Omega)$ is isometric to a subset of all Borel measures withthe 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 constructbi-Lipschitz embeddings of the set of unordered $m$-tuples in $Wb(\Omega)$ intoHilbert space. This generalises Almgren's bi-Lipschitz embedding theorem to thesetting of optimal partial transport.

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