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Metric and Geometric Relaxations of Self-Contracted Curves

Author

Listed:
  • Aris Daniilidis

    (Universidad de Chile)

  • Robert Deville

    (Université de Bordeaux 1)

  • Estibalitz Durand-Cartagena

    (UNED)

Abstract

The metric notion of a self-contracted curve (respectively, self-expanded curve, if we reverse the orientation) is hereby extended in a natural way. Two new classes of curves arise from this extension, both depending on a parameter, a specific value of which corresponds to the class of self-expanded curves. The first class is obtained via a straightforward metric generalization of the metric inequality that defines self-expandedness, while the second one is based on the (weaker) geometric notion of the so-called cone property (eel-curve). In this work, we show that these two classes are different; in particular, curves from these two classes may have different asymptotic behavior. We also study rectifiability of these curves in the Euclidean space, with emphasis in the planar case.

Suggested Citation

  • Aris Daniilidis & Robert Deville & Estibalitz Durand-Cartagena, 2019. "Metric and Geometric Relaxations of Self-Contracted Curves," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 81-109, July.
    • Handle: RePEc:spr:joptap:v:182:y:2019:i:1:d:10.1007_s10957-018-1408-0
    DOI: 10.1007/s10957-018-1408-0
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