Growth rate of Lipschitz constants for retractions between finite subset spaces
Volume 260 / 2021
Studia Mathematica 260 (2021), 317-326
MSC: Primary 54E40; Secondary 46B20, 54B20, 54C15.
DOI: 10.4064/sm200527-2-11
Published online: 10 May 2021
Abstract
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.
