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) ⊂ X(2) ⊂ ・ ・ ・. 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) → X(n − 1) for all n ≥ 2. We prove that when X is a normed space, the above sequence admits Lipschitz retractions X(n) → X, X(n) → X(2), as well as concrete retractions X(n) → 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) → X(n − 1), n ≥ 2, exist for X in more general classes of Banach spaces.