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

Abstract

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