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

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• 作者：Daniel Azagra^a ; ^{azagra@mat.ucm.es" class="auth_mail" title="E-mail the corresponding author} ; Carlos Mudarra^b ; ^{carlos.mudarra@icmat.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Convex function ; C1 ; 1C1 ; 1 function ; Whitney extension theorem
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1167-1182
• 全文大小：369 K

文摘

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