Linear programming with spheres and hemispheres of objective vectors

Given a linear program with a bounded p-dimensional feasible region let the objective vector range over an s-sphere, that is, an s-dimensional sphere centered at the origin where s does not exceed p-1. If the feasible region and the sphere are in general position with respect to each other, then the corresponding set of all optimal solutions is a topological s-sphere. Similar results are developed for unbounded feasible regions and hemispheres of objective vectors.