v2007.09.13 - Convex Optimization

42 CHAPTER 2. CONVEX GEOMETRYNow let’s move to an ambient space of three dimensions. Figure 11(c)shows a polygon rotated into three dimensions. For a line to pass through itszero-dimensional boundary (one of its vertices) tangentially, it must existin at least the two dimensions of the polygon. But for a line to passtangentially through a single arbitrarily chosen point in the relative interiorof a one-dimensional face on the boundary as illustrated, it must exist in atleast three dimensions.Figure 11(d) illustrates a solid circular pyramid (upside-down) whoseone-dimensional faces are line-segments emanating from its pointed end(its vertex). This pyramid’s boundary is constituted solely by theseone-dimensional line-segments. A line may pass through the boundarytangentially, striking only one arbitrarily chosen point relatively interior toa one-dimensional face, if it exists in at least the three-dimensional ambientspace of the pyramid.From these few examples, way deduce a general rule (without proof):A line may pass tangentially through a single arbitrarily chosen pointrelatively interior to a k-dimensional face on the boundary of a convexEuclidean body if the line exists in dimension at least equal to k+2.Now the interesting part, with regard to Figure 12 showing a boundedpolyhedron in R 3 ; call it P : A line existing in at least four dimensions isrequired in order to pass tangentially (without hitting int P) through a singlearbitrary point in the relative interior of any two-dimensional polygonal faceon the boundary of polyhedron P . Now imagine that polyhedron P is itselfa three-dimensional face of some other polyhedron in R 4 . To pass a linetangentially through polyhedron P itself, striking only one point from itsrelative interior rel int P as claimed, requires a line existing in at least fivedimensions.This rule can help determine whether there exists unique solution to aconvex optimization problem whose feasible set is an intersecting line; e.g.,the trilateration problem (5.4.2.2.4).2.1.8 intersection, sum, difference, product2.1.8.0.1 Theorem. Intersection. [46,2.3.1] [228,2]The intersection of an arbitrary collection of convex sets is convex. ⋄