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

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Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305182&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305182&_rdoc=1&_issn=0022247X&md5=5d390a911df704c61c37e443aa97be05" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$\mathbb{H}$ be a Hilbert space, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305182&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305182&_rdoc=1&_issn=0022247X&md5=c2b17ee9df3d0438abf22202f3085a67" title="Click to view the MathML source">E⊂Hclass="mathContainer hidden">class="mathCode">$E\subset \mathbb{H}$ be an arbitrary subset and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305182&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305182&_rdoc=1&_issn=0022247X&md5=685f4720c7300f9573b496b18cf52e9c" title="Click to view the MathML source">f:E→Rclass="mathContainer hidden">class="mathCode">$f:E\to \mathbb{R}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305182&_mathId=si4.gif&_user=111111111&_pii=S0022247X16305182&_rdoc=1&_issn=0022247X&md5=ba93ff62d266b200326f971a684f90eb" title="Click to view the MathML source">G:E→Hclass="mathContainer hidden">class="mathCode">$G:E\to \mathbb{H}$ be two functions. We give a necessary and sufficient condition on the pair class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305182&_mathId=si44.gif&_user=111111111&_pii=S0022247X16305182&_rdoc=1&_issn=0022247X&md5=36b7c1784c957792e049478befd85919" title="Click to view the MathML source">(f,G)class="mathContainer hidden">class="mathCode">$(f,G)$ for the existence of a convex   function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305182&_mathId=si6.gif&_user=111111111&_pii=S0022247X16305182&_rdoc=1&_issn=0022247X&md5=2a24c51a3d27fd19e4b5ef63b2328535" title="Click to view the MathML source">F∈C^1,1(H)class="mathContainer hidden">class="mathCode">$F\in C^{1,1}(\mathbb{H})$ such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305182&_mathId=si42.gif&_user=111111111&_pii=S0022247X16305182&_rdoc=1&_issn=0022247X&md5=ca54695218612c6d74a6b39427e806d7" title="Click to view the MathML source">F=fclass="mathContainer hidden">class="mathCode">$F=f$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305182&_mathId=si43.gif&_user=111111111&_pii=S0022247X16305182&_rdoc=1&_issn=0022247X&md5=78220e08b6d48edb857949418f5c1773" title="Click to view the MathML source">∇F=Gclass="mathContainer hidden">class="mathCode">$\mathrm{\nabla }F=G$ on E. We also show that, if this condition is met, F   can be taken so that class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305182&_mathId=si51.gif&_user=111111111&_pii=S0022247X16305182&_rdoc=1&_issn=0022247X&md5=9afe4ad17b1342d2d5a46751ab937bbb">class="imgLazyJSB inlineImage" height="19" width="131" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305182-si51.gif">class="mathContainer hidden">class="mathCode">$\mathrm{Lip}(\mathrm{\nabla }F)=\mathrm{Lip}(G)$. We give a geometrical application of this result, concerning interpolation of sets by boundaries of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305182&_mathId=si10.gif&_user=111111111&_pii=S0022247X16305182&_rdoc=1&_issn=0022247X&md5=c80594904b7f71cf3faa867f72f72414" title="Click to view the MathML source">C^1,1class="mathContainer hidden">class="mathCode">$C^{1,1}$ convex bodies in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305182&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305182&_rdoc=1&_issn=0022247X&md5=5d390a911df704c61c37e443aa97be05" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$\mathbb{H}$. Finally, we give a counterexample to a related question concerning smooth convex extensions of smooth convex functions with derivatives which are not uniformly continuous.