Sphere Packings, VI. Tame Graphs and Linear Programs
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AbstractThis paper is the sixth and final part in a series of papers devoted
to the proof of the Kepler conjecture, which asserts that no packing
of congruent balls in three dimensions has density greater than the
face-centered cubic packing. In a previous paper in this series, a
continuous function f on a compact space is defined, certain
points in the domain are conjectured to give the global maxima, and
the relation between this conjecture and the Kepler conjecture is
established. In this paper we consider the set of all points in the
domain for which the value of f is at least the conjectured
maximum. To each such point, we attach a planar graph. It is proved
that each such graph must be isomorphic to a tame graph, of
which there are only finitely many up to isomorphism. Linear
programming methods are then used to eliminate all possibilities,
except for three special cases treated in earlier papers:
pentahedral prisms, the face-centered cubic packing, and the
hexagonal-close packing. The results of this paper rely on long
computer calculations.
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[2] Thomas C. Hales. Some Algorithms Arising in the Proof of the Kepler Conjecture , 2002 .
[3] T. Hales. The Kepler conjecture , 1998, math/9811078.