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

Aris Daniilidis (1) , Guy David (2) , Estibalitz Durand-Cartagena (3) , Antoine Lemenant (4)

1 UAB - Departament de Matemàtiques [Barcelona]
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 Departamento de Matemática Aplicada
4 LJLL - Laboratoire Jacques-Louis Lions

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.

Mots clés: Self-contracted curve; rectifiable curve; convex foliation; secant; self-expanded curve; proximal algorithm

Domaines: Analyse classique [math.CA]

Aris Daniilidis, Guy David, Estibalitz Durand-Cartagena, Antoine Lemenant. Rectifiability of Self-contracted curves in the Euclidean space and applications. Journal of Geometric Analysis, 2015, 25 (2), pp.1211-1239. ⟨10.1007/s12220-013-9464-z⟩. ⟨hal-00751643⟩