v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
44 CHAPTER 2. CONVEX GEOMETRY(a)R 2(b)R 3(c)(d)Figure 14: Line tangential: (a) (b) to relative interior of a zero-dimensionalface in R 2 , (c) (d) to relative interior of a one-dimensional face in R 3 .
2.1. CONVEX SET 45Now let’s move to an ambient space of three dimensions. Figure 14cshows 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 14d illustrates a solid circular cone (drawn truncated) whoseone-dimensional faces are halflines emanating from its pointed end (vertex).This cone’s boundary is constituted solely by those one-dimensional halflines.A line may pass through the boundary tangentially, striking only onearbitrarily chosen point relatively interior to a one-dimensional face, if itexists in at least the three-dimensional ambient space of the cone.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 20 showing a boundedpolyhedron in R 3 ; call it P : A line existing in at least four dimensionsis required in order to pass tangentially (without hitting int P) through asingle arbitrary point in the relative interior of any two-dimensional polygonalface on the boundary of polyhedron P . Now imagine that polyhedron P isitself a three-dimensional face of some other polyhedron in R 4 . To pass aline tangentially through polyhedron P itself, striking only one point fromits relative interior rel int P as claimed, requires a line existing in at leastfive dimensions. 2.9It is not too difficult to deduce:A line may pass through a single arbitrarily chosen point interior to ak-dimensional convex Euclidean body (hitting no other interior point)if that line exists in dimension at least equal to k+1.In layman’s terms, this means: a being capable of navigating four spatialdimensions (one Euclidean dimension beyond our physical reality) could seeinside three-dimensional objects.2.9 This rule can help determine whether there exists unique solution to a convexoptimization problem whose feasible set is an intersecting line; e.g., the trilaterationproblem (5.4.2.3.5).
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44 CHAPTER 2. CONVEX GEOMETRY(a)R 2(b)R 3(c)(d)Figure 14: Line tangential: (a) (b) to relative interior of a zero-dimensionalface in R 2 , (c) (d) to relative interior of a one-dimensional face in R 3 .
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