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Self-contracted curves are gradient flows of convex functions


Authors: Estibalitz Durand-Cartagena and Antoine Lemenant
Journal: Proc. Amer. Math. Soc. 147 (2019), 2517-2531
MSC (2010): Primary 34A99; Secondary 46N10
DOI: https://doi.org/10.1090/proc/14407
Published electronically: February 14, 2019
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Abstract: In this paper we prove that any $ C^{1,\frac {1}{2}}$ curve in $ \mathbb{R}^n$ is the solution of the gradient flow equation for some $ C^1$ convex function $ f$ if and only if it is strongly self-contracted.


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Estibalitz Durand-Cartagena
Affiliation: Departamento de Matemática Aplicada, UNED. ETSI Industriales, Juan del Rosal 12, 28040 Madrid, Spain
Email: edurand@ind.uned.es

Antoine Lemenant
Affiliation: Université Paris 7 (Denis Diderot), Laboratoire Jacques Louis Lions (CNRS UMR 7598), Université Paris Diderot - Paris 7, U.F.R. de Mathématiques, Bâtiment Sophie Germain, 75205 Paris Cedex 13, France
Email: lemenant@ljll.univ-paris-diderot.fr

DOI: https://doi.org/10.1090/proc/14407
Keywords: Self-contracted curves, gradient flow equation, convex analysis, convex optimization, convex extension, analysis in metric spaces
Received by editor(s): February 21, 2018
Received by editor(s) in revised form: February 22, 2018, July 16, 2018, and August 27, 2018
Published electronically: February 14, 2019
Additional Notes: The first author was supported by the grant MTM2015-65825-P (MINECO of Spain).
The second author was supported by the research PGMO grant COCA from the Hadamard Foundation.
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2019 American Mathematical Society