OVERVIEW xiii
show that there exist only finitely many non-equivalent secondary cones (see Theo-
rem 4.13). This gives a foundation for systematic searches for thin periodic sphere
coverings. Another application is a possible search for “good” periodic quantizers.
In Section 4.3 we introduce an “equivariant G-theory” for secondary cones and,
more general, a theory of T -secondary cones, in analogy to the theory of G-perfect
and T -perfect periodic sets described in Section 3.4. As in Section 3.4 we derive
from Lemma 3.26 a finiteness result for rational t and finite symmetry groups G
(see Theorem 4.19). We put special efforts into an explicit description on how to
obtain all Delone subdivisions of lattices and periodic sets within the G- and T -
theory (see in particular Theorem 4.15 and the algorithms in Section 4.3.4). This
practicability of our results is of great importance for the applications in Chapter 5.
In Section 4.4 we introduce the secondary cones of single Delone polyhedra
and polyhedral complexes. We propose an algorithm to decide whether or not a
given simplex with integral vertices is Delone for some positive definite quadratic
form. It can possibly serve as a tool to classify lattice Delone simplices (up to
GLd(Z)-equivalence) of a given dimension. We show that the number of inequivalent
Delone simplices increases dramatically with the dimension, by constructing Delone
simplices with relative volume increasing super-exponentially with the dimension.
Previously only linear growth was known.
Chapter 5: Local analysis of coverings and applications. In the fifth
chapter we harvest the fruits of Chapter 4 and apply them to obtain several results
in context of the lattice sphere covering problem. The first two sections have a
preparatory character.
In Section 5.1 we formulate algorithms that allow to solve (in principle) the lat-
tice sphere covering and packing-covering problem in a given dimension by solving
a finite number of convex optimization problems. We give explicit descriptions of
determinant maximization and semidefinite programs. They are subject to linear
matrix inequalities expressing that the covering radius of an underlying periodic set
is bounded by some given constant (see Proposition 5.5). We describe the software
tools we developed to find local lattice covering and packing-covering optima (or at
least for generating certified bounds on their covering density or packing-covering
constant).
In Section 5.2 we provide tools for a detailed local analysis. These allow to
check for local optimality of a lattice, if necessary computationally. Using convex
optimization software, we sometimes only have certified ranges for the actual locally
optimal value. We show how this restricted information can nevertheless be used
to obtain structural informations about the actual local optimizers. We derive
quickly computable, local lower bounds for the lattice covering density and the
packing-covering constant, applicable to all lattices having a given collection of
Delone simplices. For some lattices (as seen for the Leech lattice in Section 5.5)
these bounds can be tight (for the right choice of simplices attaining the covering
radius). Hence they have the potential to prove local optimality of a lattice.
In Section 5.3 we explain how the tools from the previous two sections can be
used to find new best known lattice coverings and packing-coverings. Our tech-
niques enable us in particular to confirm all previously known results up to dimen-
sion 5. It is notable that these were previously obtained (without computer assis-
tance) in many years of work and on hundreds of published pages. Beyond the previ-
ously known, we extend the knowledge up to dimension 5 by additional information
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