7. Let P C R" be a nonempty polyhedron. The lineality space of P is the set lin( P) := recc(P) n (- recc(P)) = {d ER" : x + td E P Vx E P, VER} (a) Show that lin(P) is a linear subspace. (b) Show that Pn (lin(P) ) is a polyhedron that contains no lines, where the symbol denotes the orthogonal complement. Clarification: The orthogonal complement L- of a linear subspace L is defined as L' = {v ER" : vd = 0 Vd E lin(P) } (c) Show that P is the Minkowski sum of lin(P) and Pn (lin(P) -). (d) Conclude that any nonempty polyhedron P can be written as a Minkowski sum P = C+K +L where C is a polytope, K is a pointed polyhedral cone, and L is a linear subspace. (e) Compute sets C, K, L as above for the polyhedron P = (x ER' : x1 + x2 2 4, x1 + 2