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