Sylvester-gallai results and other contributions to combinatorial and computational geometry

The first two chapters of this thesis concern Sylvester's problem and its relatives. The first chapter extends the well known Theorem of Csima and Sawyer which states that an arrangement of n lines not all passing through a common point, which is not the Kelly-Moser arrangement of 7 lines and 3 ordinary points, must have at least 6n13 ordinary points. We show that besides the McKee arrangement of 13 lines and 6 ordinary points, there cannot be additional arrangements with 6n13 ordinary points if n is odd.Chapter Two concerns a strong form of the dual of Sylvester's problem in the affine plane. The main result of the chapter is that in an arrangement of n lines in the affine plane, not all of which are parallel, and not all of which pass through a common point, there must be at least 2n-37 affine ordinary points as long as n ≠ 6. Chapter Three describes the relationship of the affine Sylvester problem considered in Chapter Two to the problem of finding arrangements with a maximum number of wedges. In this chapter we prove results about the maximum number of wedges and extend these results to give results concerning the minimum complexity of the outer layer. Chapter Four concerns collections of points and a notion of data depth called opposite quadrant depth. The main result of this chapter states that, given n points in the plane, there must be a point in the set of opposite quadrant depth n8 , and that, moreover, there are point sets for which there is no point with opposite quadrant depth greater than n8 + O(1). Chapter Five treats the generic problem of finding the minimum cost way of covering points (clients) by disks, given suitable cost functions, different possible placements of the clients, and constraints on how one may place the disks. In some cases exact solutions are obtained, and in others (1 + ε) or constant factor approximations are achieved. In a couple of instances NP-hardness results for the exact solution are given.