An Extension Theorem for convex functions of class C1,1 on Hilbert spaces

被引：9

作者：

Azagra, Daniel ^{[1
]}

Mudarra, Carlos ^{[2
]}

机构：

[1] Univ Complutense, Fac Ciencias Matemat, Dept Anal Matemat, ICMAT CSIC UAM UC3 UCM, E-28040 Madrid, Spain

[2] ICMAT CSIC UAM UC3 UCM, Calle Nicolas Cabrera 13-15, Madrid 28049, Spain

来源：

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS | 2017年 / 446卷 / 02期

关键词：

Convex function; C-1; function; Whitney extension theorem;

D O I：

10.1016/j.jmaa.2016.09.015

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Let H be a Hilbert space, E subset of 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 is an element of C-1,C-1 (H) such that F = f and del F = G on E. We also show that, if this condition is met, F can be taken so that Lip(del F) = Lip(G). We give a geometrical application of this result, concerning interpolation of sets by boundaries of C-1,C-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. (C) 2016 Elsevier Inc. All rights reserved.

引用

页码：1167 / 1182

页数：16