On the piecewise approximation of bi-Lipschitz curves

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 $L^{\prime }$-biLipschitz, for instance this was already done with $L^{\prime }=4L$ in [Daneri-Pratelli, Lemma 5.5]. The main result of this paper is to do the same with $L^{\prime }=L+\epsilon$ (which is of course the best possible result); in the end, we generalize the result to the case of closed curves.