On Lipschitz Retraction of Finite Subsets of Normed Spaces
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866914914194423808 |
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| author | Akofor, Earnest |
| author_facet | Akofor, Earnest |
| contents | If $X$ is a metric space, then its finite subset spaces $X(n)$ form a nested sequence under natural isometric embeddings $X = X(1)\subset X(2) \subset \cdots$. It was previously established, by Kovalev when $X$ is a Hilbert space and, by Bačák and Kovalev when $X$ is a CAT(0) space, that this sequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$ for all $n\geq 2$. We prove that when $X$ is a normed space, the above sequence admits Lipschitz retractions $X(n)\rightarrow X$, $X(n)\rightarrow X(2)$, as well as concrete retractions $X(n)\rightarrow X(n-1)$ that are Lipschitz if $n=2,3$ and Hölder-continuous on bounded sets if $n>3$. We also prove that if $X$ is a geodesic metric space, then each $X(n)$ is a $2$-quasiconvex metric space. These results are relevant to certain questions in the aforementioned previous work which asked whether Lipschitz retractions $X(n)\rightarrow X(n-1)$, $n\geq 2$, exist for $X$ in more general classes of Banach spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1811_00603 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | On Lipschitz Retraction of Finite Subsets of Normed Spaces Akofor, Earnest Functional Analysis General Topology Metric Geometry 54E40, 46B20 (Primary), 54B20, 54C15, 54C25 (Secondary) If $X$ is a metric space, then its finite subset spaces $X(n)$ form a nested sequence under natural isometric embeddings $X = X(1)\subset X(2) \subset \cdots$. It was previously established, by Kovalev when $X$ is a Hilbert space and, by Bačák and Kovalev when $X$ is a CAT(0) space, that this sequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$ for all $n\geq 2$. We prove that when $X$ is a normed space, the above sequence admits Lipschitz retractions $X(n)\rightarrow X$, $X(n)\rightarrow X(2)$, as well as concrete retractions $X(n)\rightarrow X(n-1)$ that are Lipschitz if $n=2,3$ and Hölder-continuous on bounded sets if $n>3$. We also prove that if $X$ is a geodesic metric space, then each $X(n)$ is a $2$-quasiconvex metric space. These results are relevant to certain questions in the aforementioned previous work which asked whether Lipschitz retractions $X(n)\rightarrow X(n-1)$, $n\geq 2$, exist for $X$ in more general classes of Banach spaces. |
| title | On Lipschitz Retraction of Finite Subsets of Normed Spaces |
| topic | Functional Analysis General Topology Metric Geometry 54E40, 46B20 (Primary), 54B20, 54C15, 54C25 (Secondary) |
| url | https://arxiv.org/abs/1811.00603 |