Iterative processes generating dense point sets

Abstract

The central problem of the thesis is the question of denseness of certain sets of points in the plane. All of the following results are joint with A. Bezdek. We consider point sets P, not a subset of a line, having the property that for every three noncollinear points in P, a specific triangle center (incenter (IC), circumcenter (CC), orthocenter (OC) resp.) is also in the set P. In Chapter 2, we generalize and solve the CC problem in higher dimensions. We prove that if a point set P in the n-dimensional Euclidean space has the property that for each simplex of P the circumcenter of the simplex also belongs to P, then it is dense in the whole space. Chapter 3 contains the solution of the OC problem in the plane, essentially proving that P is either a dense point set of the plane or it is a subset of a rectangular hyperbola. In the latter case, it is either a dense subset or it is a special discrete subset of a rectangular hyperbola, for which we give both an algebraic and a geometric characterization.