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Article

Self-contracted curves have finite length

Journal of London Mathematical Society. 2017. Vol. 96. No. 2. P. 455-481.
Степанов Е. О., Teplitskaya Y.

A curve  in a metric space  equipped with the distance , where  is a (possibly unbounded) interval, is called self-contracted, if for any triple of instances of time  with  one has . We prove that if  is a finite-dimensional normed space with an arbitrary norm, the trace of  is bounded, then  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].