On the piecewise approximation of bi-Lipschitz curves

TL;DR

This paper presents an optimal method for uniformly approximating bi-Lipschitz curves with piecewise linear functions, achieving nearly the best possible bi-Lipschitz constant, and extends the results to closed curves.

Contribution

The paper improves existing approximation results by achieving an approximation with bi-Lipschitz constant L+ε, which is optimal, and generalizes the approach to closed curves.

Findings

Achieved approximation with bi-Lipschitz constant L+ε

Extended approximation method to closed curves

Improved upon previous approximation bounds

Abstract

In this paper we deal with the task of uniformly approximating an $L$-biLipschitz curve by means of piecewise linear ones. This is rather simple if one is satisfied to have approximating functions which are $L'$-biLipschitz, for instance this was already done with $L'= 4L$ in [Daneri-Pratelli, Lemma 5.5]. The main result of this paper is to do the same with $L'=L+ \varepsilon$ (which is of course the best possible result); in the end, we generalize the result to the case of closed curves.