Before revision:  In this work, we extend the theorem to general polyhedral cones and polyhedra , thereby refining Minkowski's and Caratheodory 's theorems  in polyhedral geometry.
After revision:  In this work, we extend the theorem , first to "subspace cones" and then to general polyhedral cones and polyhedra , thereby refining Minkowski's and Caratheodory 's theorems in polyhedral geometry.