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

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