On the structure of the inverse-feasible region of a linear program

Given a set of feasible solutions X to a linear program, we study the set of objectives that make X optimal, known as the inverse-feasible region. We show the relationship between the dimension of a face of a polyhedron and the dimension of the corresponding inverse-feasible region, which leads to necessary and sufficient conditions of the extreme, boundary, and inner points of a linear program. We also characterize the set of objectives that render a given solution uniquely optimal.