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

TL;DR

This paper proves that bounded self-contracted curves in Euclidean spaces have finite length, extending previous planar results to higher dimensions without requiring continuity, and explores applications in dynamical systems and optimization.

Contribution

It generalizes the finite length property of self-contracted curves to all finite-dimensional Euclidean spaces without continuity assumptions.

Findings

Bounded self-contracted curves have finite length in any finite-dimensional Euclidean space.

Applications include regularity of solutions in nonsmooth convex systems.

Proximal sequences converge under any parameter choice, simplifying optimization proofs.

Abstract

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