Rectifiability of Self-Contracted Curves in the Euclidean Space and Applications

A. Daniilidis, G. David, E. Durand-Cartagena, A. Lemenant

Research output: Contribution to journalArticleResearchpeer-review

13 Citations (Scopus)

Abstract

© 2013, Mathematica Josephina, Inc. 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 et al. (J. Math. Pures Appl. 94:183–199, 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. Dedic. 38:211–227, 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 a 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.
Original languageEnglish
Pages (from-to)1211-1239
JournalJournal of Geometric Analysis
Volume25
Issue number2
DOIs
Publication statusPublished - 1 Jan 2015

Keywords

  • Convex foliation
  • Proximal algorithm
  • Rectifiable curve
  • Secant
  • Self-contracted curve
  • Self-expanded curve

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