Linear programming with spheres and hemispheres of objective vectors (Q1176803)

Consider the linear program \(\max\{cx\mid x\in P\}\) where \(P\) is a linearly constrained feasible region. Let \(X(P,c)\) denote the set of all optimal solutions. For any set \(C\) of possible \(c\)'s, define \(X(P,C)=\bigcup_{c\in C}X(P,c)\). The authors determine topological possibilities of \(X(P,C)\) for \(C\) a sphere or hemisphere. For this purpose the feasible region \(P\) is projected onto the linear span of objective vectors where the situation is simplified, then conclusions obtained in the projected space are transferred back to the original region.