Bi-Lipschitz embeddings of the space of unordered \(m\)-tuples with a partial transportation metric

Let \(\Omega \subset {\mathbb {R}}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be non-empty, open and proper. This paper is concerned with \(Wb_p(\Omega )\) $W{b}_p(\mathrm{\Omega })$ , the space of p-integrable Borel measures on \(\Omega \) $\mathrm{\Omega }$ equipped with the partial transportation metric introduced by Figalli and Gigli that allows the creation and destruction of mass on \(\partial \Omega \) $\partial \mathrm{\Omega }$ . Alternatively, we show that \(Wb_p(\Omega )\) $W{b}_p(\mathrm{\Omega })$ is isometric to a subset of Borel measures with the ordinary Wasserstein distance, on the one point completion of \(\Omega \) $\mathrm{\Omega }$ equipped with the shortcut metric \(\begin{aligned} \delta (x,y)= \min \{\Vert x-y\Vert , {\text {dist}}(x,\partial \Omega )+{\text {dist}}(y,\partial \Omega )\}. \end{aligned}\) $$\begin{array}{c}\hfill \delta (x,y)=min\{\Vert x-y\Vert ,\text{dist}(x,\partial \mathrm{\Omega })+\text{dist}(y,\partial \mathrm{\Omega })\}.\end{array}$$ In this article we construct bi-Lipschitz embeddings of the set of unordered m-tuples in \(Wb_p(\Omega )\) $W{b}_p(\mathrm{\Omega })$ into Hilbert space. This generalises Almgren’s bi-Lipschitz embedding theorem to the setting of optimal partial transport.