Some problems related to the polytope illumination

This thesis is a collection of results concerning the Illumination Conjecture and the combi- natorial structure of C-polyhedra. These results were developed both independently and in collaboration with my Supervisor, Dr. Károly Bezdek and a fellow graduate student, Cameron Strachan. The Illumination Conjecture concerns convex bodies in Euclidean space Ed . A direction u in the origin-centred unit sphere S^{d−1} illuminates a boundary point x of a convex body K, if the ray with direction u, that starts at x, has a nonempty intersection with the interior of K. A convex body K is illuminated by directions u_1 , . . . , u_n ∈ S^{d−1} if every boundary point of K is illuminated by some direction u_i , 1 ≤ i ≤ n. Illumination number of a convex body K, denoted by Ill(K), is the smallest number of directions that illuminate K. The Illumination Conjecture states that the illumination number Ill(K) of a d-dimensional convex body K ⊂ E^d is at most 2^d , and Ill(K) < 2^d if K is not an image of a d-cube under some linear transformation. In this thesis, we prove the Illumination Conjecture for the origin-symmetric cap bodies of a ball in E^3 , E^4 , and E^d , d ≥ 19. We also prove the upper bound Ill(K) ≤ 4d for the unconditionally symmetric cap bodies in an arbitrary dimension d ≥ 5. Furthermore, we obtain the upper bound Ill(K) ≤ 12 for the cap bodies of a ball in E^3 . Next, we prove the Illumination Conjecture for the primitive polytopes. A bounded intersection of halfspaces is a primitive polytope, if omitting any halfspace renders the intersection unbounded. A C-polyhedron P ⊂ E^d is a reduced, d-dimensional intersection of homothets of a smooth, strictly convex body C ⊂ E^d . A fa-C-t of P is a connected smooth subset of bd P that is not contained in another smooth connected subset of bd P . We prove that any C-polyhedron, that is the intersection of two homothets, has exactly two fa-C-ts. Next, we prove that a C-polygon P ⊂ E^2 , that is an intersection of n translates of C, has exactly n fa-C-ts. We also prove that a C-polygon P ⊂ E^2 , that is an intersection of n homothets of a smooth strictly convex body C ⊂ E^2 , has at most 2n − 2 fa-C-ts.