The Illumination of Symmetric Spiky Balls and Cap Bodies; and a note on the Vertex Classification of Planar C-polygons

Abstract

This thesis handles two problems in the field of convex geometry. The first, is the problem of illuminating the boundary of Euclidean shapes known as spiky balls and cap bodies in various dimensions. Specifically we consider illuminating the cases where a spiky ball is 2-illuminable, and the cases where a cap body is symmetric, either centrally or unconditionally. The second problem we investigate is how complex the boundary structure of a C-polygon can be, which is a finite intersection of homothets of a particular convex domain C; and how this boundary structure depends on C.