The $K$-hull of a compact set $Asubsetmathbb{R}^d$, where $Ksubset mathbb{R}^d$ is a fixed compact convex body, is the intersection of all translates of $K$ that contain $A$. A set is called $K$-strongly convex if it coincides with its $K$-hull. We propose a general approach to the analysis of facial structure of $K$-strongly convex sets, similar to the well developed theory for polytopes, by introducing the notion of $k$-dimensional faces, for all $k=0,dots,d-1$. We then apply our theory in the case when $A=Xi_n$ is a sample of $n$ points picked uniformly at random from $K$. We show that in this case the set of $xinmathbb{R}^d$ such that $x+K$ contains the sample $Xi_n$, upon multiplying by $n$, converges in distribution to the zero cell of a certain Poisson hyperplane tessellation. From this results we deduce convergence in distribution of the corresponding $f$-vector of the $K$-hull of $Xi_n$ to a certain limiting random vector, without any normalisation, and also the convergence of all moments of the $f$-vector.
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