Periodica Mathematica Hungarica
Volume/Issue:
Volume 55: Issue 1
Incenter iterations in 3-space
Authors: Gergely Ambrus and András Bezdek
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  • 1 University College London Department of Mathematics Gower Street London WC1E 6BT UK
  • 2 University of Szeged Bolyai Institute Aradi vértanúk tere 1 H-6720 Szeged Hungary
  • 3 Auburn University Department of Mathematics and Statistics 221 Parker Hall Auburn AL 36849 USA
  • 4 MTA Rényi Institute Reáltanoda u. 13-15 Budapest Hungary
DOI:
https://doi.org/10.1007/s10998-007-3113-0
Page Count:
113–119
Publication Date:
01 Sep 2007
Online Publication Date:
05 Sep 2007
Article Category:
Research Article
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Abstract  

Consider a 3-dimensional point set

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{P}$$ \end{document}
which contains the incenters of all the nondegenerate tetrahedra with vertices from
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{P}$$ \end{document}
. In this paper we prove that then
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{P}$$ \end{document}
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.

Cover Periodica Mathematica Hungarica
Periodica Mathematica Hungarica
Print ISSN:
0031-5303
Online ISSN:
1588-2829
Keywords:
52A35; dense point set; incenter; iterated process

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