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• Author or Editor: András Bezdek
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# 3rd Geometry Festival: An International Conference on Packings, Coverings and Tilings

Periodica Mathematica Hungarica
Authors: András Bezdek and Károly Bezdek
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# Circle Covering With a Margin

Periodica Mathematica Hungarica
Authors: András Bezdek and Włodzimierz Kuperberg
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# Incenter iterations in 3-space

Periodica Mathematica Hungarica
Authors: Gergely Ambrus and András Bezdek

## 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.
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