Daniilidis, ArisDeville, Robert2024-12-032024-12-032018-10-13Daniilidis, 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/s10957-018-1408-01573-2878https://doi.org/10.1007/s10957-018-1408-0https://hdl.handle.net/20.500.14468/24678The registered version of this article, first published in Journal of Optimization Theory and Applications, is available online at the publisher's website: Springer Nature, https://doi.org/10.1007/s10957-018-1408-0La versión registrada de este artículo, publicado por primera vez en Journal of Optimization Theory and Applications, está disponible en línea en el sitio web del editor: Springer Nature, https://doi.org/10.1007/s10957-018-1408-0The 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.eninfo:eu-repo/semantics/openAccess12 Matemáticasartículoself-contracted curveself-expanded curverectifiabilitylengthλ-curveλ-cone property