v2010.10.26 - Convex Optimization

2.1. CONVEX SET 392.1.6 empty set versus empty interiorEmptiness ∅ of a set is handled differently than interior in the classicalliterature. It is common for a nonempty convex set to have empty interior;e.g., paper in the real world:An ordinary flat sheet of paper is a nonempty convex set having emptyinterior in R 3 but nonempty interior relative to its affine hull.2.1.6.1 relative interiorAlthough it is always possible to pass to a smaller ambient Euclidean spacewhere a nonempty set acquires an interior [26,II.2.3], we prefer the qualifierrelative which is the conventional fix to this ambiguous terminology. 2.6 Sowe distinguish interior from relative interior throughout: [330] [371] [358]Classical interior int C is defined as a union of points: x is an interiorpoint of C ⊆ R n if there exists an open ball of dimension n and nonzeroradius centered at x that is contained in C .Relative interior rel int C of a convex set C ⊆ R n is interior relative toits affine hull. 2.7Thus defined, it is common (though confusing) for int C the interior of C tobe empty while its relative interior is not: this happens whenever dimensionof its affine hull is less than dimension of the ambient space (dim aff C < n ;e.g., were C paper) or in the exception when C is a single point; [258,2.2.1]rel int{x} aff{x} = {x} , int{x} = ∅ , x∈ R n (12)In any case, closure of the relative interior of a convex set C always yieldsclosure of the set itself;rel int C = C (13)Closure is invariant to translation. If C is convex then rel int C and C areconvex. [199, p.24] If C has nonempty interior, thenrel int C = int C (14)2.6 Superfluous mingling of terms as in relatively nonempty set would be an unfortunateconsequence. From the opposite perspective, some authors use the term full orfull-dimensional to describe a set having nonempty interior.2.7 Likewise for relative boundary (2.1.7.2), although relative closure is superfluous.[199,A.2.1]