Calculus of Variations and Geometric Measure Theory

E. Stepanov - Y. Teplitskaya

Self-contracted curves have finite length

created by stepanov on 23 Aug 2016
modified on 13 Oct 2017


Published Paper

Inserted: 23 aug 2016
Last Updated: 13 oct 2017

Journal: J. London Math. Soc.
Volume: 96
Number: 2
Pages: 455–481
Year: 2017
Doi: 10.1112/jlms.12068

ArXiv: 1707.04922v1 PDF


A curve $\theta$: $I\to E$ in a metric space $E$ equipped with the distance $d$, where $I\subset \mathbb{R}$ is a (possibly unbounded) interval, is called self-contracted, if for any triple of instances of time $\{t_i\}_{i=1}^3\subset I$ with $t_1\leq t_2\leq t_3$ one has $d(\theta(t_3),\theta(t_2))\leq d(\theta(t_3),\theta(t_1))$. We prove that if $E$ is a finite-dimensional normed space with an arbitrary norm, the trace of $\theta$ is bounded, then $\theta$ has finite length, i.e. is rectifiable, thus answering positively the question raised in A. Lemenant. Rectifiability of non Euclidean self-contracted curves. arXiv: 1604.02673 math, 2016.

Keywords: self-contracted curve, anisotropic norm, rectifiable curve, steepest descent, gradient system