On the extension of bi-Lipschitz mappings

TL;DR

This paper proves that closed semialgebraic sets of dimension k can be bi-Lipschitz embedded into R^n for n ≥ 2k+1, with uniqueness up to bi-Lipschitz homeomorphism when n ≥ 2k+2.

Contribution

It establishes conditions for bi-Lipschitz embeddings of semialgebraic sets into Euclidean spaces and proves their uniqueness in higher dimensions.

Findings

Existence of bi-Lipschitz embeddings for n ≥ 2k+1.

Uniqueness of embeddings when n ≥ 2k+2.

Embeddings are semialgebraic and bi-Lipschitz.

Abstract

Let $X$ be a closed semialgebraic set of dimension $k.$ If $n\ge 2k+1$, then there is a bi-Lipschitz and semialgebraic embedding of $X$ into $\Bbb R^n.$ Moreover, if $n \ge 2k+2$, then this embedding is unique (up to a bi-Lipschitz and semialgebraic homeomorphism of $\Bbb R^n.$