Lipschitz retraction of finite subsets of Hilbert spaces

TL;DR

This paper proves that for finite subsets of a Hilbert space, there exist Lipschitz retractions from the space of size n to size n-1, establishing a structured relationship within these nested finite subset spaces.

Contribution

It establishes the existence of Lipschitz retractions between finite subset spaces in Hilbert spaces, a novel structural insight.

Findings

Lipschitz retractions exist for finite subset spaces in Hilbert spaces.

The nested sequence of finite subset spaces admits Lipschitz retractions.

This result enhances understanding of the geometric structure of finite subsets in Hilbert spaces.

Abstract

Finite subset spaces of a metric space $X$ form a nested sequence under natural isometric embeddings $X=X(1)\subset X(2)\subset\dots$. We prove that this sequence admits Lipschitz retractions $X(n)\to X(n-1)$ when $X$ is a Hilbert space.