Metric and geometric relaxations of self-contracted curves

TL;DR

This paper introduces metric and geometric relaxations of self-contracted curves, establishing conditions under which such curves have finite length in Euclidean spaces and highlighting differences between planar and higher-dimensional cases.

Contribution

It extends the concept of self-contractedness to metric and geometric notions called λ-curves and λ-cone property, analyzing their properties and length bounds.

Findings

Bounded λ-curves have finite length in  for 

Existence of infinite-length curves satisfying λ-cone property in  for   with  

All bounded planar curves with the λ-cone property have finite length

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.