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arXiv · 1707.04922

Self-contracted curves have finite length

Eugene Stepanov·Yana Teplitskaya
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Abstract

A curve $θ$: $I\to E$ in a metric space $E$ equipped with the distance $d$, where $I\subset \R$ is a (possibly unbounded) interval, is called self-contracted, if for any triple of instances of time $\{t_i\}_{i=1}^3\subset I$ with $t_1\leq t_2\leq t_3$ one has $d(θ(t_3),θ(t_2))\leq d(θ(t_3),θ(t_1))$. 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~\cite{Lemenant16sc-rectif}.

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Eugene Stepanov, Yana Teplitskaya. 2017-07-16. Self-contracted curves have finite length. https://arxiv.org/abs/1707.04922

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