On the piecewise approximation of bi-Lipschitz curves (Q1700656)

Summary: In this paper we deal with the task of uniformly approximating an \(L\)-bi-Lipschitz curve by means of piecewise linear ones. This is rather simple if one is satisfied to have approximating functions which are \(L^\prime\)-bi-Lipschitz, for instance, this was already done with \(L^\prime=4L\) in [\textit{S. Daneri} and \textit{A. Pratelli}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, No. 3, 567--589 (2014; Zbl 1348.37071), Lemma 5.5]. The main result of this paper is to do the same with \(L^\prime=L+\varepsilon\) (which is, of course, the best possible result); in the end, we generalize the result to the case of closed curves.