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