论文标题
Lipschitz常数的生长速率，用于有限子集之间的缩回
Growth rate of Lipschitz constants for retractions between finite subset spaces
论文作者
论文摘要
对于任何度量空间$ x $，$ x $的有限子集空间提供一系列等距嵌入$ x = x（1）\ subset x（2）\ subset \ cdots $。 Lipschitz缩回的存在$ r_n \ colon x（n）\ to x（n-1）$取决于以微妙的方式的几何形状。当$ x $是hadamard空间或有限维规范的空间时，已知这种缩回存在。但是，即使在这些情况下，尚不清楚序列$ \ {r_n \} $是否可以统一地lipschitz。我们通过证明$ \ operatatorName {lip}（r_n）$必须在$ n $中生长出负面答案，而$ x $是标准的空间或hadamard空间。
For any metric space $X$, finite subset spaces of $X$ provide a sequence of isometric embeddings $X=X(1)\subset X(2)\subset\cdots$. The existence of Lipschitz retractions $r_n\colon X(n)\to X(n-1)$ depends on the geometry of $X$ in a subtle way. Such retractions are known to exist when $X$ is an Hadamard space or a finite-dimensional normed space. But even in these cases it was unknown whether the sequence $\{r_n\}$ can be uniformly Lipschitz. We give a negative answer by proving that $\operatorname{Lip}(r_n)$ must grow with $n$ when $X$ is a normed space or an Hadamard space.
