Metric and geometric relaxations of self-contracted curves

Abstract

Self-contractedness (or self-expandedness, depending on the orientation) is hereby extended in two natural ways giving rise, for any $\lambda \in [-1,1)$, 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 $\lambda \in [-1,1/d)$ bounded $\lambda$-curves have finite length. For $\lambda \ge 1/\sqrt{5}$ 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.

20 pages, 3 figures