Self‐contracted curves have finite length

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 = 1 3 ⊂ I witht 1 ⩽ t 2 ⩽ t 3one has d ( θ ( t 3 ) , θ ( t 2 ) ) ⩽ 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, 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].