Metric and geometric relaxations of self-contracted curves

2018/02/26 by Aris Daniilidis, Daniilidis, Aris, Robert Deville +3
Mathematics · #FOS: Mathematics #Metric Geometry (math.MG) #math.MG

paper · pdf · doi:10.48550/arxiv.1802.09637

20 pages, 3 figures

arxiv created 2018/02/26 · arxiv updated 2018/02/28

Abstract

Self-contractedness (or self-expandedness, depending on the orientation) is hereby extended in two natural ways giving rise, for any λ∈\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 ℝ^d it is established that for λ∈\lbrack-1,1/d) bounded λ-curves have finite length. For λ≥ 1/√(5) it is always possible to construct bounded curves of infinite length in ℝ³ which do satisfy the λ-cone property. This can never happen in ℝ² though: it is shown that all bounded planar curves with the λ-cone property have finite length.

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