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.
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