v2007.11.26 - Convex Optimization

2.8. CONE BOUNDARY 95For a proper cone in vector space R n with n ≥ 2, we can say more:{exposed directions} = {extreme directions} (159)It follows from Lemma 2.8.0.0.1 for any pointed closed convex cone, thereis one-to-one correspondence of one-dimensional exposed faces with exposeddirections; id est, there is no one-dimensional exposed face that is not a raybase 0.The pointed closed convex cone EDM 2 , for example, is a ray inisomorphic subspace R whose relative boundary (2.6.1.3.1) is the origin.The conventionally exposed directions of EDM 2 constitute the empty set∅ ⊂ {extreme direction}. This cone has one extreme direction belonging toits relative interior; an idiosyncrasy of dimension 1.2.8.2.1 Connection between boundary and extremes2.8.2.1.1 Theorem. Exposed. [232,18.7] (confer2.8.1.1.1)Any closed convex set C containing no lines (and whose dimension is atleast 2) can be expressed as the closure of the convex hull of its exposedpoints and exposed rays.⋄From Theorem 2.8.1.1.1,rel∂C = C \ rel int C (141)= conv{exposed points and exposed rays} \ rel int C= conv{extreme points and extreme rays} \ rel int C⎫⎪⎬⎪⎭(160)Thus each and every extreme point of a convex set (that is not a point)resides on its relative boundary, while each and every extreme direction of aconvex set (that is not a halfline and contains no line) resides on its relativeboundary because extreme points and directions of such respective sets donot belong to the relative interior by definition.The relationship between extreme sets and the relative boundary actuallygoes deeper: Any face F of convex set C (that is not C itself) belongs torel ∂ C , so dim F < dim C . [232,18.1.3]2.8.2.2 Converse caveatIt is inconsequent to presume that each and every extreme point and directionis necessarily exposed, as might be erroneously inferred from the conventional