Calculus of Variations and Geometric Measure Theory

A. Pratelli - E. Radici

On the piecewise approximation of bi-Lipschitz curves

created by pratelli on 08 May 2015
modified by radici on 28 Oct 2022


Published Paper

Inserted: 8 may 2015
Last Updated: 28 oct 2022

Journal: Rend. Sem. Mat. Univ. Padova
Volume: 138
Pages: 1-37
Year: 2017

ArXiv: 1505.06510 PDF


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.