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It is hereby established that, in Euclidean spaces of finite dimension,bounded self-contracted curves have finite length. This extends the main resultof Daniilidis, Ley, and Sabourau (J. Math. Pures Appl. 2010) concerningcontinuous planar self-contracted curves to any dimension, and dispensesentirely with the continuity requirement. The proof borrows heavily from ageometric idea of Manselli and Pucci (Geom. Dedicata 1991) employed for thestudy of regular enough curves, and can be seen as a nonsmooth adaptation ofthe latter, albeit a nontrivial one. Applications to continuous and discretedynamical systems are discussed: continuous self-contracted curves appear asgeneralized solutions of nonsmooth convex foliation systems, recovering ahidden regularity after reparameterization, as consequence of our main result.In the discrete case, proximal sequences (obtained through implicitdiscretization of a gradient system) give rise to polygonal self-contractedcurves. This yields a straightforward proof for the convergence of the exactproximal algorithm, under any choice of parameters.

Rectifiability of Self-contracted curves in the Euclidean space and applications