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Let $S$ be a convex hypersurface (the boundary of a closed convex set $V$with nonempty interior) in $\mathbb{R}^n$. We prove that $S$ contains no linesif and only if for every open set $U\supset S$ there exists a real-analyticconvex hypersurface $S_{U} \subset U\cap \textrm{int}(V) $. We also show that$S$ contains no rays if and only if for every open set $U\supset S$ thereexists a real-analytic convex hypersurface $S_{U}\subset U\setminus V$.Moreover, in both cases, $S_U$ can be taken strongly convex. We also establishsimilar results for convex functions defined on open convex subsets of$\mathbb{R}^n$, completely characterizing the class of convex functions thatcan be approximated in the $C^0$-fine topology by smooth convex functions fromabove or from below. We also provide similar results for $C^1$-fineapproximations

Inner and outer smooth approximation of convex hypersurfaces. When is it possible?