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Growth rate of Lipschitz constants for retractions between finite subset spaces
- Source :
- Studia Math. 260 (2021), no. 3, 317-326
- Publication Year :
- 2020
-
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.
- Subjects :
- Mathematics - Metric Geometry
Primary 54E40, Secondary 46B20, 54B20, 54C15
Subjects
Details
- Database :
- arXiv
- Journal :
- Studia Math. 260 (2021), no. 3, 317-326
- Publication Type :
- Report
- Accession number :
- edsarx.2005.13579
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.4064/sm200527-2-11