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Let $X$ denote $\mathbb{R}^n$ or, more generally, a Hilbert space. Given anarbitrary subset $C$ of $X$ and a collection $\mathcal{H}$ of affinehyperplanes of $X$ such that every $H\in\mathcal{H}$ passes through some point$x_{H}\in C$, and $C=\{x_H : H\in\mathcal{H}\}$, what conditions are necessaryand sufficient for the existence of a $C^{1,1}$ convex hypersurface $S$ in $X$such that $H$ is tangent to $S$ at $x_H$ for every $H\in\mathcal{H}$? In thispaper we give an answer to this question. We also provide solutions to similarproblems for convex hypersurfaces of class $C^{1, \omega}$ in Hilbert spaces,and for convex hypersurfaces of class $C^{1, \alpha}$ in superreflexive Banachspaces having equivalent norms with moduli of smoothness of power type$1+\alpha$, $\alpha\in (0, 1].$

Prescribing tangent hyperplanes to $C^{1,1}$ and $C^{1,\omega}$ convex hypersurfaces in Hilbert and superreflexive Banach spaces