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Azagra Rueda, Daniel and Mudarra, C.
(2017)
*An Extension Theorem for convex functions of class C1,1 on Hilbert spaces.*
Journal of Mathematical Analysis and Applications, 446
(2).
pp. 1167-1182.
ISSN 0022-247X

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Official URL: http://www.sciencedirect.com/science/article/pii/S0022247X16305182

## Abstract

Let H be a Hilbert space, E⊂H be an arbitrary subset and f:E→R, G:E→H be two functions. We give a necessary and sufficient condition on the pair (f,G) for the existence of a convex function F∈C1,1(H) such that F=f and ∇F=G on E. We also show that, if this condition is met, F can be taken so that Lip(∇F)=Lip(G). We give a geometrical application of this result, concerning interpolation of sets by boundaries of C1,1 convex bodies in H. Finally, we give a counterexample to a related question concerning smooth convex extensions of smooth convex functions with derivatives which are not uniformly continuous.

Item Type: | Article |
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Uncontrolled Keywords: | C1,1 function; Convex function; Whitney extension theorem |

Subjects: | Sciences > Mathematics > Mathematical analysis |

ID Code: | 41483 |

Deposited On: | 27 Feb 2017 11:52 |

Last Modified: | 27 Feb 2017 12:20 |

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