We introduce the notion of g-polytope and that of g-capping of a closed convex cone, which, respectively, generalize the notions of weakly compact set and that of a cappable cone. Then we develop both a theory of g-polytopes, which are closed convex sets containing no rays, and a corresponding theory of generalized capping of convex cones, where we require that the cut cone be a g-polytope, instead of a convex weakly compact set. We will also prove a KM-like theorem for the special class of g-polytopes, which occur in the range space theory of polyhedra.