v2006.03.09 - Convex Optimization

136 CHAPTER 2. CONVEX GEOMETRY2.12 Convex polyhedraEvery polyhedron, such as the convex hull (71) of a bounded list X , canbe expressed as the solution set of a finite system of linear equalities andinequalities, and vice versa. [61,2.2]2.12.0.0.1 Definition. Convex polyhedra, halfspace-description.[38,2.2.4] A convex polyhedron is the intersection of a finite number ofhalfspaces and hyperplanes;P = {y | Ay ≽ b, Cy = d} ⊆ R n (233)where coefficients A and C generally denote matrices. Each row of C is avector normal to a hyperplane, while each row of A is a vector inward-normalto a hyperplane partially bounding a halfspace.△By the halfspaces theorem in2.4.1.1.1, a polyhedron thus described is aclosed convex set having possibly empty interior; e.g., Figure 11. Convexpolyhedra 2.38 are finite-dimensional comprising all affine sets (2.3.1),polyhedral cones, line segments, rays, halfspaces, convex polygons, solids[138, def.104/6, p.343], polychora, polytopes, 2.39 etcetera.It follows from definition (233) by exposure that each face of a convexpolyhedron is a convex polyhedron.The projection of any polyhedron on a subspace remains a polyhedron.More generally, the image of a polyhedron under any linear transformationis a polyhedron. [17,I.9]When b and d in (233) are 0, the resultant is a polyhedral cone. Theset of all polyhedral cones is a subset of convex cones:2.38 We consider only convex polyhedra throughout, but acknowledge the existence ofconcave polyhedra. [244, Kepler-Poinsot Solid]2.39 Some authors distinguish bounded polyhedra via the designation polytope. [61,2.2]