Locally $C^{1,1}$ convex extensions of $1$-jets

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Let E be an arbitrary subset of \mathbb{R}^n , and let f\colon E\to\mathbb{R} , G\colon E\to\mathbb{R}^n be given functions. We provide necessary and sufficient conditions for the existence of a convex function F\in C^{1,1}_{\textrm{loc}}(\mathbb{R}^n) such that F=f and \nabla F=G on E . We give a useful explicit formula for such an extension F , and a variant of our main result for the class C^{1, \omega}_{\textrm{loc}} , where \omega is a modulus of continuity. We also present two applications of these results, concerning how to find C^{1,1}_{\textrm{loc}} convex hypersurfaces with prescribed tangent hyperplanes on a given subset of \mathbb{R}^n , and some explicit formulas for (not necessarily convex) C^{1,1}_{\textrm{loc}} extensions of 1 -jets.