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 Ba\v{c}\'{a}k 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 H\"older-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.