# Metric and geometric relaxations of self-contracted curves

created by durandcar on 27 Feb 2018

[BibTeX]

Preprint

Inserted: 27 feb 2018
Last Updated: 27 feb 2018

Year: 2018

Abstract:

Self-contractedness (or self-expandedness, depending on the orientation) is hereby extended in two natural ways giving rise, for any $\lambda\in\lbrack-1,1)$, to the metric notion of $\lambda$-curve and the (weaker) geometric notion of $\lambda$-cone property ($\lambda$-eel). In the Euclidean space $\mathbb{R}^{d}$ it is established that for $\lambda\in\lbrack-1,1/d)$ bounded $\lambda$-curves have finite length. For $\lambda\geq 1/\sqrt{5}$ it is always possible to construct bounded curves of infinite length in ${\mathbb{R}}^{3}$ which do satisfy the $\lambda$-cone property. This can never happen in ${\mathbb{R}}^{2}$ though: it is shown that all bounded planar curves with the $\lambda$-cone property have finite length.