*Accepted Paper*

**Inserted:** 23 aug 2016

**Last Updated:** 12 jul 2017

**Journal:** J. London Math. Soc.

**Year:** 2016

**Abstract:**

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

**Download:**