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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Daniilidis, A., Ley, O., Sabourau, S.: Asymptotic behavior of self-contracted planar curves and gradient orbits of convex functions. J. Math. Pures Appl. 94, 183–199 (2010)
Manselli, P., Pucci, C.: Maximum length of steepest descent curves for quasi-convex functions. Geom. Dedicata 38, 211–227 (1991)
Daniilidis, A., Drusvyatskiy, D., Lewis, A.S.: Orbits of geometric descent. Can. Math. Bull. 58, 44–50 (2015)
Longinetti, M., Manselli, P., Venturi, A.: On steepest descent curves for quasi convex families in \({\mathbb{R}}^{n}\). Math. Nachr. 288, 420–442 (2015)
Daniilidis, A., David, G., Durand-Cartagena, E., Lemenant, A.: Rectifiability of self-contracted curves in the Euclidean space and applications. J. Geom. Anal. 25, 1211–1239 (2015)
Ohta, S.: Self-contracted curves in CAT(0)-spaces and their rectifiability (Preprint). arXiv:1711.09284
Zolotov, V.: Sets with small angles in self-contracted curves (Preprint). arXiv:1804.00234
Daniilidis, A., Deville, R., Durand-Cartagena, E., Rifford, L.: Self-contracted curves in Riemannian manifolds. J. Math. Anal. Appl. 457, 1333–1352 (2018)
Lemenant, A.: Rectifiability of non Euclidean planar self-contracted curves. Conflu. Math. 8, 23–38 (2016)
Stepanov, E., Teplitskaya, Y.: Self-contracted curves have finite length. J. Lond. Math. Soc. 96, 455–481 (2017)
Rockafellar, R.T., Wets, R.: Variational Analysis. Grundlehren der Mathematischen, Wissenschaften, vol. 317. Springer, Berlin (1998)
Aichholzer, O., Aurenhammer, F., Icking, C., Klein, R., Langetepe, E., Rote, G.: Generalized Self-Approaching Curves. In: 14th European Workshop on Computational Geometry CG98 (Barcelona). Discrete Appl. Math. 109(1–2), 3–24 (2001)
Acknowledgements
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Daniilidis, A., Deville, R. & Durand-Cartagena, E. Metric and Geometric Relaxations of Self-Contracted Curves. J Optim Theory Appl 182, 81–109 (2019). https://doi.org/10.1007/s10957-018-1408-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-018-1408-0