Symmetric products of the line: embeddings and retractions

TL;DR

This paper proves that symmetric products of the line can be embedded into high-dimensional Euclidean spaces and are absolute Lipschitz retracts, advancing understanding of their geometric structure.

Contribution

It establishes that all symmetric products of the line are absolute Lipschitz retracts and can be bi-Lipschitz embedded into high-dimensional Euclidean spaces.

Findings

Symmetric products of the line are absolute Lipschitz retracts.

They admit bi-Lipschitz embeddings into high-dimensional Euclidean spaces.

The results enhance understanding of the geometric properties of symmetric products.

Abstract

The n-th symmetric product of a metric space is the set of its nonempty subsets with cardinality at most n, equipped with the Hausdorff metric. We prove that every symmetric product of the line is an absolute Lipschitz retract and admits a bi-Lipschitz embedding into a Euclidean space of sufficiently high dimension.