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  • 1 Bolyai Institute, University of Szeged & Department of Mathematics and Statistics, Auburn University 1 Aradi vértanúk tere, H-6720 Szeged, Hungary & 221 Parker Hall, Auburn, AL 36849, U.S.A 1 Aradi vértanúk tere, H-6720 Szeged, Hungary & 221 Parker Hall, Auburn, AL 36849, U.S.A
  • 2 Department of Geometry, Bolyai Institute University of Szeged 1 Aradi vértanúk tere, H-6720 Szeged, Hungary 1 Aradi vértanúk tere, H-6720 Szeged, Hungary
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Summary  

We prove that the mininum surface area of a Voronoi cell in a unit ball packing in \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} \({\mathbb E}^3\) \end{document} is at least \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} \(16.1977\) \end{document}. This result provides further support for the Strong Dodecahedral Conjecture according to which the minimum surface area of a Voronoi cell in a \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} \(3\) \end{document}-dimensional unit ball packing is at least as large as the surface area of a regular dodecahedron of inradius \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} \(1\) \end{document}, which is about \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} \(16.6508\ldots\,\) \end{document}.

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