On Lipschitz Retraction of Finite Subsets of Normed Spaces

Abstract

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 $X(n)\to X(n-1)$ for all $n\ge 2$. We prove that when $X$ is a normed space, the above sequence admits Lipschitz retractions $X(n)\to X$, $X(n)\to X(2)$, as well as concrete retractions $X(n)\to X(n-1)$ 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 $X(n)\to X(n-1)$, $n\ge 2$, exist for $X$ in more general classes of Banach spaces.

20 pages, Isr. J. Math. (2019). "$\gamma$ is injective" added in Lemma 6.6(ii), Published in Israel Journal of Mathematics

References in corpus (2)

• Lipschitz retraction of finite subsets of Hilbert spaces
• Lipschitz retractions in Hadamard spaces via gradient flow semigroups

Cited by in corpus (3)

• Lipschitz clustering in metric spaces
• Anti-absorbing ternary operations on metric spaces
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