Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions

Aris Daniilidis, Olivier Ley, Stéphane Sabourau

Research output: Contribution to journalArticleResearchpeer-review

18 Citations (Scopus)

Abstract

We hereby introduce and study the notion of self-contracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded self-contracted planar curves have a finite length. We also give an example of a convex function defined in the plane whose gradient orbits spiral infinitely many times around the unique minimizer of the function. © 2010 Elsevier Masson SAS.
Original languageEnglish
Pages (from-to)183-199
JournalJournal des Mathematiques Pures et Appliquees
Volume94
Issue number2
DOIs
Publication statusPublished - 1 Aug 2010

Keywords

  • Łojasiewicz inequality
  • Convex foliation
  • Convex function
  • Gradient trajectory
  • Planar dynamical system

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