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arXiv · 2005.13579

Growth rate of Lipschitz constants for retractions between finite subset spaces

Earnest Akofor·Leonid V. Kovalev
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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.

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Earnest Akofor, Leonid V. Kovalev. 2020-05-27. Growth rate of Lipschitz constants for retractions between finite subset spaces. https://doi.org/10.4064/sm200527-2-11

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