Partitioning problems in discrete and computational geometry

Many interesting problems in Discrete and Computational Geometry involve partitioning. A main question is whether a given set, or sets, may be separated into parts satisfying certain properties. Sometime we also need to find an efficient way to do it—in other words an algorithm. In this thesis, we discuss several of problems and results of this kind. First we give a combinatorial proof of the existence and the uniqueness of the generalized ham-sandwich cut for well separated point sets in Rd, that have the weak general position property. The combinatorial proof allows us to derive an O( n(log n)d -3) running time algorithm to find a generalized cut for d given well separated point sets in R d. A second problem concerns the 6-way partition of a given convex set in R2 using 3 lines. We reopen an old question and show that when the direction of one line is fixed, there is unique partition such that 6 regions have the same area. In the Voronoi game two players A and B each play n points in a compact set S in Rd. They obtain scores equal to the total volume of the Voronoi cells of their points. There are one round and alternating versions of the game. We give the player A’s best strategy to minimize the area of player B’s one (first) Voronoi cell.