Immobilization of convex bodies in $R^n$

We extend to arbitrary finite $n$ the notion of immobilization of a convex body $O$ in $R^n$ by a finite set of points $P$ in the boundary of $O$. Because of its importance for this problem, necessary and sufficient conditions are found for the immobilization of an $n$-simplex. A fairly complete geometric description of these conditions is given: as $n$ increases from $n = 2$, some qualitative difference in the nature of the sets $P$ emerges.