{
  "id": "1211.3227",
  "version": "v1",
  "published": "2012-11-14T07:35:09.000Z",
  "updated": "2012-11-14T07:35:09.000Z",
  "title": "Rectifiability of Self-contracted curves in the Euclidean space and applications",
  "authors": [
    "Aris Daniilidis",
    "Guy David",
    "Estibalitz Durand-Cartagena",
    "Antoine Lemenant"
  ],
  "categories": [
    "math.CA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-14T07:35:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "euclidean space",
      "continuous planar self-contracted curves",
      "applications",
      "rectifiability",
      "nonsmooth convex foliation systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.3227D"
    }
  }
}