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

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 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.

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Acknowledgements

The first author acknowledges support of the grants MTM2011-29064-C01 (Spain) and FONDECYT 1130176 (Chile) and thanks Jerome Bolte and Joel Benoist for useful discussions. The third author is partially supported by grant MTM2009-07848 (Spain). The second and fourth authors are partially supported by the ANR project GEOMETRYA (France). Part of this work has been realized during a research stay of the third author at the Université Paris Diderot (Paris 7) and Laboratory Jacques Louis Lions. The stay was supported by the program “Research in Paris” offered by the Ville de Paris (Mairie de Paris). This author thanks the host institution and Ville de Paris for its hospitality.

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Correspondence to G. David.

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Research of A.D. supported by the grant MTM2011-29064-C01 (Spain) and by the FONDECYT Regular Grant No. 1130176 (Chile).

Communicated by Steven G. Krantz.

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Daniilidis, A., David, G., Durand-Cartagena, E. et al. Rectifiability of Self-Contracted Curves in the Euclidean Space and Applications. J Geom Anal 25, 1211–1239 (2015). https://doi.org/10.1007/s12220-013-9464-z

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Keywords

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

Mathematics Subject Classification

  • 53A04
  • 37N40
  • 49J52
  • 49J53
  • 52A10
  • 65K10