Growth rate of Lipschitz constants for retractions between finite subset spaces

TL;DR

This paper investigates the Lipschitz constants of retractions between finite subset spaces of metric spaces, showing that these constants must grow with the subset size in normed and Hadamard spaces, indicating limitations in uniform Lipschitz retractions.

Contribution

It proves that Lipschitz retractions between finite subset spaces cannot have uniformly bounded Lipschitz constants in normed and Hadamard spaces, resolving an open question.

Findings

Lipschitz constants grow with subset size in normed spaces

Lipschitz constants grow with subset size in Hadamard spaces

Uniform Lipschitz retractions do not exist in these spaces

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.