v2010.10.26 - Convex Optimization

46 CHAPTER 2. CONVEX GEOMETRY2.1.7.2 Relative boundaryThe classical definition of boundary of a set C presumes nonempty interior:∂ C = C \ int C (17)More suitable to study of convex sets is the relative boundary; defined[199,A.2.1.2]rel ∂ C C \ rel int C (24)boundary relative to affine hull of C .In the exception when C is a single point {x} , (12)rel∂{x} = {x}\{x} = ∅ , x∈ R n (25)A bounded convex polyhedron (2.3.2,2.12.0.0.1) in subspace R , forexample, has boundary constructed from two points, in R 2 from atleast three line segments, in R 3 from convex polygons, while a convexpolychoron (a bounded polyhedron in R 4 [373]) has boundary constructedfrom three-dimensional convex polyhedra. A halfspace is partially boundedby a hyperplane; its interior therefore excludes that hyperplane. An affineset has no relative boundary.2.1.8 intersection, sum, difference, product2.1.8.0.1 Theorem. Intersection. [307,2, thm.6.5]Intersection of an arbitrary collection of convex sets {C i } is convex. For afinite collection of N sets, a necessarily nonempty intersection of relativeinterior ⋂ Ni=1 rel int C i = rel int ⋂ Ni=1 C i equals relative interior of intersection.And for a possibly infinite collection, ⋂ C i = ⋂ C i .⋄In converse this theorem is implicitly false in so far as a convex set canbe formed by the intersection of sets that are not. Unions of convex sets aregenerally not convex. [199, p.22]Vector sum of two convex sets C 1 and C 2 is convex [199, p.24]C 1 + C 2 = {x + y | x ∈ C 1 , y ∈ C 2 } (26)but not necessarily closed unless at least one set is closed and bounded.