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

Let $$ be non-empty, open and proper. This paper is concerned with $$ , the space of p-integrable Borel measures on $$ equipped with the partial transportation metric introduced by Figalli and Gigli that allows the creation and destruction of mass on $$ . Alternatively, we show that $$ is isometric to a subset of Borel measures with the ordinary Wasserstein distance, on the one point completion of $$ equipped with the shortcut metric In this article we construct bi-Lipschitz embeddings of the set of unordered m-tuples in $$ into Hilbert space. This generalises Almgren's bi-Lipschitz embedding theorem to the setting of optimal partial transport.