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ON A STRONG VERSION OF THE KEPLER CONJECTURE

Part of:Designs and configurationsPolytopes and polyhedraDiscrete geometry

Published online by Cambridge University Press:  01 August 2012

Károly Bezdek
Károly Bezdek*
Affiliation:
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW Calgary, Canada T2N 1N4Department of Mathematics, University of Pannonia, Veszprém, HungaryInstitute of Mathematics, Eötvös University, Budapest, Hungary (email: bezdek@math.ucalgary.ca)

Abstract

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We raise and investigate the following problem which one can regard as a very close relative of the densest sphere packing problem. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells? In particular, we prove that the average surface area in question is always at least $\frac {24}{\sqrt {3}}=13.8564\ldots .$

MSC classification

Secondary:05B40: Packing and covering05B45: Tessellation and tiling problems52B60: Isoperimetric problems for polytopes52C17: Packing and covering in $n$ dimensions52C22: Tilings in $n$ dimensions

Information

Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013, pp. 23 - 30
Copyright
Copyright © University College London 2012

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