v2009.01.01 - Convex Optimization

42 CHAPTER 2. CONVEX GEOMETRY
Now let’s move to an ambient space of three dimensions. Figure 13(c)
shows a polygon rotated into three dimensions. For a line to pass through its
zero-dimensional boundary (one of its vertices) tangentially, it must exist
in at least the two dimensions of the polygon. But for a line to pass
tangentially through a single arbitrarily chosen point in the relative interior
of a one-dimensional face on the boundary as illustrated, it must exist in at
least three dimensions.
Figure 13(d) illustrates a solid circular pyramid (upside-down) whose
one-dimensional faces are line-segments emanating from its pointed end
(its vertex). This pyramid’s boundary is constituted solely by these
one-dimensional line-segments. A line may pass through the boundary
tangentially, striking only one arbitrarily chosen point relatively interior to
a one-dimensional face, if it exists in at least the three-dimensional ambient
space of the pyramid.
From these few examples, way deduce a general rule (without proof):
A line may pass tangentially through a single arbitrarily chosen point
relatively interior to a k-dimensional face on the boundary of a convex
Euclidean body if the line exists in dimension at least equal to k+2.
Now the interesting part, with regard to Figure 16 showing a bounded
polyhedron in R 3 ; call it P : A line existing in at least four dimensions is
required in order to pass tangentially (without hitting int P) through a single
arbitrary point in the relative interior of any two-dimensional polygonal face
on the boundary of polyhedron P . Now imagine that polyhedron P is itself
a three-dimensional face of some other polyhedron in R 4 . To pass a line
tangentially through polyhedron P itself, striking only one point from its
relative interior rel int P as claimed, requires a line existing in at least five
dimensions.
This rule can help determine whether there exists unique solution to a
convex 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, product
2.1.8.0.1 Theorem. Intersection. [53,2.3.1] [266,2]
The intersection of an arbitrary collection of convex sets is convex. ⋄