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Resumen:
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Let C be a subset of ?n (not necessarily convex), f : C ? R be a function and G : C ? ?n be a uniformly continuous function, with modulus of continuity ?. We provide a necessary and sufficient condition on f, G for the existence of a convex function F ? CC1?(?n) such that F = f on C and ?F = G on C, with a good control of the modulus of continuity of ?F in terms of that of G. On the other hand, assuming that C is compact, we also solve a similar problem for the class of C1 convex functions on ?n, with a good control of the Lipschitz constants of the extensions (namely, Lip(F) ? ?G??). Finally, we give a geometrical application concerning interpolation of compact subsets K of ?n by boundaries of C1 or C1,1 convex bodies with prescribed outer normals on K.
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