Self-contracted curves have finite length

2017/07/16 by Eugene Stepanov, Stepanov, Eugene, Yana Teplitskaya +1
Mathematics · #FOS: Mathematics #Metric Geometry (math.MG) #math.MG

paper · pdf · doi:10.48550/arxiv.1707.04922

27 pages

arxiv created 2017/07/16 · arxiv updated 2017/07/18

Abstract

A curve θ: I→ E in a metric space E equipped with the distance d, where I⊂ \R is a (possibly unbounded) interval, is called self-contracted, if for any triple of instances of time \t_i\_i=1³⊂ I with t₁≤ t₂≤ t₃ one has d(θ(t₃),θ(t₂))≤ d(θ(t₃),θ(t₁)). We prove that if E is a finite-dimensional normed space with an arbitrary norm, the trace of θ is bounded, then θ has finite length, i.e. is rectifiable, thus answering positively the question raised in~\citeLemenant16sc-rectif.

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