v2010.10.26 - Convex Optimization

2.12. CONVEX POLYHEDRA 147all dimensions except n ; their meanings become equivalent as we saw inFigure 20 (discussed in2.6.1.2). In other words, each and every face of anypolyhedral set (except the set itself) can be exposed by a hyperplane, andvice versa; e.g., Figure 24.Lewis [246,6] [215,2.3.4] claims nonempty extreme proper subsets andthe exposed subsets coincide for S n + ; id est, each and every face of the positivesemidefinite cone, whose dimension is less than dimension of the cone, isexposed. A more general discussion of cones having this property can befound in [341]; e.g., Lorentz cone (178) [20,II.A].2.12 Convex polyhedraEvery polyhedron, such as the convex hull (86) of a bounded list X , canbe expressed as the solution set of a finite system of linear equalities andinequalities, and vice versa. [113,2.2]2.12.0.0.1 Definition. Convex polyhedra, halfspace-description.A convex polyhedron is the intersection of a finite number of halfspaces andhyperplanes;P = {y | Ay ≽ b, Cy = d} ⊆ R n (286)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 isa closed convex set possibly not full-dimensional; e.g., Figure 20. Convexpolyhedra 2.53 are finite-dimensional comprising all affine sets (2.3.1),polyhedral cones, line segments, rays, halfspaces, convex polygons, solids[223, def.104/6 p.343], polychora, polytopes, 2.54 etcetera.It follows from definition (286) by exposure that each face of a convexpolyhedron is a convex polyhedron.Projection of any polyhedron on a subspace remains a polyhedron. Moregenerally, image and inverse image of a convex polyhedron under any linear2.53 We consider only convex polyhedra throughout, but acknowledge the existence ofconcave polyhedra. [373, Kepler-Poinsot Solid]2.54 Some authors distinguish bounded polyhedra via the designation polytope. [113,2.2]