  Let $X$ denote $\mathbb{R}^n$ or, more generally, a Hilbert space. Given an
arbitrary subset $C$ of $X$ and a collection $\mathcal{H}$ of affine
hyperplanes 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 necessary
and 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 this
paper we give an answer to this question. We also provide solutions to similar
problems for convex hypersurfaces of class $C^{1, \omega}$ in Hilbert spaces,
and for convex hypersurfaces of class $C^{1, \alpha}$ in superreflexive Banach
spaces having equivalent norms with moduli of smoothness of power type
$1+\alpha$, $\alpha\in (0, 1].$
