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Do correction terms with the required
properties exist? The evidence suggests
that they exist in abundance. The first
correction term was proposed by L.
Fejes T\'oth in 1953, but his clusters
were much larger than than those used
here. His clusters contain so many
balls that fcc-compatibility has never
been established. Nevertheless, his
proposal represents a significance
advance, because it gave the first
evidence that the Kepler conjecture
could be solved through an optimization
problem in a finite number of
variables. He proposed in 1964 that
computers might be used to determine
the minimum. Thus, the general strategy
of a proof was set.
The correction terms f are based on a
careful study of the local geometry of
sphere packings. Fejes T\'oth's
correction term f(p) has the form
with q running
over all centers of balls in the
cluster p within a fixed distance of
the center of the cluster. The term
v(q) is the volume of a truncated (a
different truncation constant is used
in this approach) Voronoi cell centered
at q. The constants a(p,q) sum to
zero, åq a(p,q) = 0, for all
clusters p in a packing. This
zero-sum condition leads to a
cancellation of the terms in å f(p), and hence to the transience of
f. This correction term illustrates
the general correction term strategy:
f is constructed as sums of volumes
that are added at p and subtracted
again at q. The sum åf(p)
behaves like the telescoping series
1-1+1-1+¼, and each term of the
sum is swallowed by the next in its
path. Transience results from this
cancellation of terms.
Of course, there is no need for the
volumes that are shuffled between p
and q to be Voronoi cells. One of
many other possibilities is known as
the Delaunay decomposition. In that
decomposition, an edge is drawn between
two centers of balls if their Voronoi
cells share a face. These edges form
simplices, known as Delaunay simplices,
and the simplices partition
space.
When asked to name the most difficult
part of the proof of the Kepler
conjecture, I answer without hesitation
that it was the design of the
decomposition of space implicit in f.
I had worked with Voronoi cells without
success and had also tried Delaunay
simplices. Both approaches became
complicated beyond my ability to
understand them. My progress stopped.
Finally, one day in November 1994, I
realized how to combine the two
approaches into a hybrid decomposition
that retained the best features of
each. From that day on, I never
waivered in my confidence that the
Kepler conjecture would eventually be
solved by the hybrid
approach.
Hybrid correction terms are extremely
flexible and easy to construct, and
soon Sam Ferguson and I realized that
every time we encountered difficulties
in solving the minimization problem, we
could adjust f to skirt the
difficulty. The function f became
more complicated, but with each change
we cut months - or even years - from
our work. This incessant fiddling was
unpopular with my colleagues. Every
time I presented my work in progress at
a conference, I was minimizing a
different function. Even worse, the
correction function in my early papers
differs from the one in the final
papers, and this required me to go back
and patch the old papers. The
correction function did not become
fixed until it came time for Sam to
defend his thesis, and we finally felt
obligated to stop tampering with it.
However, if I were to revise the proof
to produce a simpler one, the first
thing I would do would be to change the
correction function once again. It is
the key to a simple proof.
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