论文标题
无序的$ m $ tuples的Bi-Lipschitz嵌入，带有部分运输度量
Bi-Lipschitz embeddings of the space of unordered $m$-tuples with a partial transportation metric
论文作者
论文摘要
令$ω\ subset \ mathbb {r}^n $是非空的，开放和正确的。考虑$ wb（ω）$，这是有限的borel措施的$ω$，配备了Figalli和Gigli引入的部分运输度量的$ω$，允许在$ \ partialω$上创建和破坏质量。同等地，我们表明，$ wb（ω）$与普通的Wasserstein距离的所有Borel措施的子集相同，配备了$ω$，配备了$ω$，配备了快捷方式\ [Δ（x，x，y）= \ \ min \ min \ min \ { ω）+\ operatorName {dist}（y，\partialΩ）\}。\ \]在本文中，我们构造了$ m $ tuplaces的bi-lipschitz嵌入$ wb（ω）$中的$ m $ tuplace in in Hilbert Space。这将Almgren的Bi-Lipschitz嵌入到最佳部分运输的设置中。
Let $Ω\subset \mathbb{R}^n$ be non-empty, open and proper. Consider $Wb(Ω)$, the space of finite Borel measures on $Ω$ equipped with the partial transportation metric introduced by Figalli and Gigli that allows the creation and destruction of mass on $\partial Ω$. Equivalently, we show that $Wb(Ω)$ is isometric to a subset of all Borel measures with the ordinary Wasserstein distance, on the one point completion of $Ω$ equipped with the shortcut metric \[δ(x,y)= \min\{\|x-y\|, \operatorname{dist}(x,\partial Ω)+\operatorname{dist}(y,\partialΩ)\}.\] In this article we construct bi-Lipschitz embeddings of the set of unordered $m$-tuples in $Wb(Ω)$ into Hilbert space. This generalises Almgren's bi-Lipschitz embedding theorem to the setting of optimal partial transport.
