|dc.description.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.||en_US