On iterative processes generating dense point sets
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ITERATIVE PROCESSES GENERATING DENSE POINT SETS G. Ambrus1,2 (Szeged, Auburn) and A. Bezdek2,3 (Auburn, Budapest) 1
Bolyai Institute, University of Szeged, 1 Aradi v´ertan´ uk tere, H-6720 Szeged, Hungary
2
3
Department of Mathematics and Statistics, 221 Parker Hall, Auburn University, Auburn, AL 36849, U.S.A.
MTA R´enyi Institute, 13–15 Re´ altanoda u., Budapest, Hungary (Received: November 20, 2005; Accepted: February 24, 2006)
Abstract The central problem of this paper is the question of denseness of those planar point sets P, not a subset of a line, which have 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. The IC and CC versions were settled before. First we generalize and solve the CC problem in higher dimensions. Then we solve 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.
1. Introduction History and terminology. There are several results in the literature which deal with iterative processes in the plane. A typical problem starts with the description of a geometric construction, which when applied to an initial point set generates additional points and thus expands it. The problem usually is to prove that repeated expansions lead to an everywhere dense point set in the plane. Mathematics subject classification number: 52A35. Key words and phrases: dense point set, orthocenter, incenter, circumcenter. The first author was supported by OTKA Grant T049398, the second author was supported by OTKA Grants T038397 and T043520. 0031-5303/2006/$20.00 c Akad´ emiai Kiad´ o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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g. ambrus and a. bezdek
For example K. Bezdek and J. Pach [2] answering a question of L. Fejes T´ oth proved√ that if one starts with two points (whose distance is ≤ 2 and different from 1 and 3 ) then repeated use of the construction “add to the figure all intersection points of those unit circles whose centers belong to the existing points set” leads to a dense point set. Recently D. Ismailescu and R. Radoi˘ci´c [4] (and earlier B. Gr¨ unbaum) showed that the repeated use of the construction “add to the figure all the intersection points of lines which connect pairs of already existing points” also leads to a dense point set (with the exception of a few particular starting configurations). Ismailescu also suggested to study similar problems where one uses the construction “add the circumcenters (incenters resp.) of all nondegenerate triangles formed by the existing points”. It was proved in [5] that in case of the circumcenters the iterative process always leads to a dense point set of the plane, and in case of the incenters it always leads to a dense point set in the convex hull of the initial point set. Higher dimensional circumcenter problem. In Section 2
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