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Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions

Abstract : We hereby introduce and study the notion of self-contracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded self-contracted planar curves have a finite length. We also give an example of a convex function defined in the plane whose gradient orbits spiral infinitely many times around the unique minimum of the function.
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Contributor : Olivier Ley <>
Submitted on : Friday, August 29, 2008 - 4:12:09 PM
Last modification on : Friday, April 12, 2019 - 4:46:03 PM
Document(s) archivé(s) le : Thursday, June 3, 2010 - 7:12:17 PM

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Aris Daniilidis, Olivier Ley, Stéphane Sabourau. Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions. Journal de Mathématiques Pures et Appliquées, Elsevier, 2010, 94 (2), pp.183-199. ⟨10.1016/j.matpur.2010.03.007⟩. ⟨hal-00315675⟩

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