Metric and Geometric Relaxations of Self-Contracted Curves

Aris Daniilidis (), Robert Deville () and Estibalitz Durand-Cartagena ()
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Aris Daniilidis: Universidad de Chile
Robert Deville: Université de Bordeaux 1
Estibalitz Durand-Cartagena: UNED

Journal of Optimization Theory and Applications, 2019, vol. 182, issue 1, No 5, 109 pages

Abstract: 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.

Keywords: Self-contracted curve; Self-expanded curve; Rectifiability; Length; $$\lambda $$ λ -curve; $$\lambda $$ λ -cone; 53A04; 52A10 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10957-018-1408-0

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