Metric and Geometric Relaxations of Self-Contracted Curves
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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.
KeywordsSelf-contracted curve Self-expanded curve Rectifiability Length \(\lambda \)-curve \(\lambda \)-cone
Mathematics Subject Classification53A04 52A10
The authors wish to thank K. Kurdyka, L. Rifford and F. Sanz for useful discussions and the two referees for their careful reading and suggestions. A. Daniilidis is supported by the Grants: BASAL AFB170001, FONDECYT 1171854, ECOS/CONICYT C14E06, REDES/CONICYT 150040 (Chile) and MTM2014-59179-C2-1-P (MINECO of Spain and ERDF of EU). R. Deville is supported by the Grants: ECOS/SUD C14E06 (France) and REDES/CONICYT-150040 (Chile). E. Durand-Cartagena is supported by the Grants MTM2015-65825-P (MINECO of Spain) and 2018-MAT14 (ETSI Industriales, UNED).