Periodica Mathematica Hungarica

, Volume 55, Issue 1, pp 113–119 | Cite as

Incenter iterations in 3-space

  • Gergely Ambrus
  • András Bezdek


Consider a 3-dimensional point set \(\mathcal{P}\) which contains the incenters of all the nondegenerate tetrahedra with vertices from \(\mathcal{P}\). In this paper we prove that then \(\mathcal{P}\) is dense in its convex hull. This settles the last unsolved variation in a sequence of similar questions initiated by D. Ismailescu, where he required to include other simplex centers, e.g. the orthocenters or the circumcenters. Our method allows us to generalize the planar incenter problem, showing that the denseness follows from a much weaker assumption for planar point sets.

Key words and phrases

dense point set incenter iterated process 

Mathematics subject classification number



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. Ambrus and A. Bezdek, On iterative processes generating dense point sets, Periodica Mathematica Hungarica, 53 (2006), 27–44.CrossRefGoogle Scholar
  2. [2]
    A. Bezdek and T. Bisztriczky, Incenter iterations in the plane and on the sphere, to appear.Google Scholar
  3. [3]
    D. Ismailescu and R. Radoicić, A dense planar point set from itarated line intersections, Computational Geometry, 27 (2004), 257–267.zbMATHCrossRefGoogle Scholar
  4. [4]
    M. Iorio, D. Ismailescu, R. Radoicić and M. Silva, On point sets containing their triangle centers, Rev. Roumaine Math. Pures Appl., 50 (2005), 677–693.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • Gergely Ambrus
    • 1
    • 2
  • András Bezdek
    • 3
    • 4
  1. 1.Department of MathematicsUniversity College LondonLondonUK
  2. 2.Bolyai InstituteUniversity of SzegedSzegedHungary
  3. 3.Department of Mathematics and StatisticsAuburn UniversityAuburnUSA
  4. 4.MTA Rényi InstituteBudapestHungary

Personalised recommendations