Show the following: (i) The intersection of an affine subspace with a polyhedron is a polyhedron....

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Show the following: (i) The intersection of an affine subspace with a polyhedron is a polyhedron. (ii) Let P1, P2 be two polytopes. Define C = polytope. {x+yx P1, y E P2}. Show that C is a (iii) The intersection of two polyhedra is a polyhedron. Thus, show that the intersection of a polyhedron with a polytope is a polytope. (iv) Let TR Rd be a linear map. Show that if PCR is a polytope, then T(P) is a polytope in Rd. [You may use the Minkowski-Weyl theorem without proof.] Show the following: (i) The intersection of an affine subspace with a polyhedron is a polyhedron. (ii) Let P1, P2 be two polytopes. Define C = polytope. {x+yx P1, y E P2}. Show that C is a (iii) The intersection of two polyhedra is a polyhedron. Thus, show that the intersection of a polyhedron with a polytope is a polytope. (iv) Let TR Rd be a linear map. Show that if PCR is a polytope, then T(P) is a polytope in Rd. [You may use the Minkowski-Weyl theorem without proof.]