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Explicit formulas for $ C^{1,1}$ Glaeser-Whitney extensions of $ 1$-Taylor fields in Hilbert spaces


Authors: Aris Daniilidis, Mounir Haddou, Erwan Le Gruyer and Olivier Ley
Journal: Proc. Amer. Math. Soc. 146 (2018), 4487-4495
MSC (2010): Primary 54C20; Secondary 52A41, 26B05, 26B25, 58C25
DOI: https://doi.org/10.1090/proc/14012
Published electronically: July 13, 2018
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Abstract | References | Similar Articles | Additional Information

Abstract: We give a simple alternative proof for the $ C^{1,1}$-convex extension problem which has been introduced and studied by D. Azagra and C. Mudarra (2017). As an application, we obtain an easy constructive proof for the Glaeser-Whitney problem of $ C^{1,1}$ extensions on a Hilbert space. In both cases we provide explicit formulae for the extensions. For the Glaeser-Whitney problem the obtained extension is almost minimal, that is, minimal up to a multiplicative factor in the sense of Le Gruyer (2009).


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Aris Daniilidis
Affiliation: DIM–CMM, UMI CNRS 2807, Beauchef 851 (Torre Norte, piso 5), Universidad de Chile, Santiago de Chile
Email: arisd@dim.uchile.cl

Mounir Haddou
Affiliation: IRMAR, INSA Rennes, CNRS UMR 6625, 20 avenue des Buttes de Coesmes, F-35708 Rennes, France
Email: mounir.haddou@insa-rennes.fr

Erwan Le Gruyer
Affiliation: IRMAR, INSA Rennes, CNRS UMR 6625, 20 avenue des Buttes de Coesmes, F-35708 Rennes, France
Email: erwan.le-gruyer@insa-rennes.fr

Olivier Ley
Affiliation: IRMAR, INSA Rennes, CNRS UMR 6625, 20 avenue des Buttes de Coesmes, F-35708 Rennes, France
Email: olivier.ley@insa-rennes.fr

DOI: https://doi.org/10.1090/proc/14012
Keywords: Whitney extension problem, convex extension, sup-inf convolution, semiconvex function.
Received by editor(s): June 12, 2017
Received by editor(s) in revised form: October 21, 2017
Published electronically: July 13, 2018
Additional Notes: This research was supported by the grants: BASAL PFB-03 (Chile), FONDECYT 1171854 (Chile) and MTM2014-59179-C2-1-P (MINECO of Spain and ERDF of EU)
Communicated by: Yuan Xu
Article copyright: © Copyright 2018 American Mathematical Society

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