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If $X$ is a metric space, then its finite subset spaces $X(n)$ form a nestedsequence under natural isometric embeddings $X = X(1)\subset X(2) \subset\cdots$. It was previously established, by Kovalev when $X$ is a Hilbert spaceand, by Ba\v{c}\'{a}k and Kovalev when $X$ is a CAT(0) space, that thissequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$ for all $n\geq2$. We prove that when $X$ is a normed space, the above sequence admitsLipschitz retractions $X(n)\rightarrow X$, $X(n)\rightarrow X(2)$, as well asconcrete retractions $X(n)\rightarrow X(n-1)$ that are Lipschitz if $n=2,3$ andH\"older-continuous on bounded sets if $n>3$. We also prove that if $X$ is ageodesic metric space, then each $X(n)$ is a $2$-quasiconvex metric space.These results are relevant to certain questions in the aforementioned previouswork which asked whether Lipschitz retractions $X(n)\rightarrow X(n-1)$, $n\geq2$, exist for $X$ in more general classes of Banach spaces.

On Lipschitz Retraction of Finite Subsets of Normed Spaces