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Article

Self-contracted curves have finite length

Journal of London Mathematical Society. 2017. Vol. 96. No. 2. P. 455-481.
Eugene Stepanov, Teplitskaya Y.

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 {ti}3i=1⊂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 ofθ is bounded, then θ has finite length, that is, is rectifiable, thus answering positively the question raised in Lemenant's paper [‘Rectifiability of non Euclidean planar self‐contracted curves’, Confluentes Math. 8 (2016) 23–38].