A convex hypersurface $\mathscr F$ in a Riemannian space $M^m$ is part of the boundary of an $m$-dimensional locally convex set. It is established that there exists an intrinsic metric of such a hypersurface $\mathscr F$ and it has curvature which is bounded below in the sense of A. D. Aleksandrov; curves with bounded variation of rotation in $\mathscr F$ are shortest paths in $M^m$. For surfaces in $R^m$ these facts are well known; however, the constructions leading to them are in large part inapplicable to spaces $M^m$. Hence approximations to $\mathscr F$ by smooth equidistant (not necessarily convex) ones and normal polygonal paths, introduced (in the case of $R^3$) by Yu. F. Borisov are used.