BiLipschitz embeddings of the space of unordered $m$tuples with a partial transportation metric
Abstract
Let $\Omega\subset \mathbb{R}^n$ be nonempty, 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 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 \[\delta(x,y)= \min\{\xy\, \operatorname{dist}(x,\partial \Omega)+\operatorname{dist}(y,\partial\Omega)\}.\] In this article we construct biLipschitz embeddings of the set of unordered $m$tuples in $Wb(\Omega)$ into Hilbert space. This generalises Almgren's biLipschitz embedding theorem to the setting of optimal partial transport.
 Publication:

arXiv eprints
 Pub Date:
 December 2022
 DOI:
 10.48550/arXiv.2212.01280
 arXiv:
 arXiv:2212.01280
 Bibcode:
 2022arXiv221201280B
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  Classical Analysis and ODEs