xiv OVERVIEW

on all local optima. Although the lattice covering and the lattice packing-covering

problem have both not been solved in dimension 6 so far, we describe conjecturally

optimal lattices obtained by heuristic methods (see Section 5.3.3). Both lattices

have the property that their Delone triangulation refines the Delone subdivision of

the lattice E6. ∗ As a first step in direction of a proof for global optimality, we prove

computationally by a branch-and-bound method that the conjectural optimal cov-

ering lattice is the unique optimum among lattices having this refinement property

(see Theorem 5.28). In many cases it is a problem to obtain exact coordinates for

the new lattices. Exemplarily we show how to obtain exact coordinates for the

new conjectural optimal 6-dimensional packing-covering lattice. We describe a new

conjecturally optimal 7-dimensional covering lattice whose corresponding positive

definite quadratic forms are (to our surprise) even rational (see Section 5.3.5). We

describe how we obtained new, currently best known covering lattices in dimensions

d ≥ 9. These computations are based on our new T -secondary cone theory.

In Section 5.4 we apply the theory of T -secondary cones to a problem in al-

gebraic number theory: the classification of all totally real thin algebraic number

fields. Our classification is based on a list of 17 candidates, previously given by

Bayer-Fluckiger and Nebe. By proving lower bounds on the covering density of

positive definite quadratic forms in an associated linear subspaces T we exclude

three of the candidates. This finishes the classification.

In Section 5.5 we take a closer look at two of the most exceptional lattices (in

low dimensions): The root lattice E8 and the Leech lattice. We show that both

lattices are rigid, meaning they are uniquely determined by the type of their Delone

subdivision. By applying our local lower bounds we show that the Leech lattice

is a local lattice covering and packing-covering optimum. This shows in particular

the existence of rigid, locally optimal lattice coverings, which aﬃrmatively answers

a long standing open question of Dickson (1968). The same method can not be

applied to the E8 root lattice. In fact, a local analysis reveals that it is not even a

locally optimal covering lattice. In connection with a proof of this fact, we find a

new currently best known 8-dimensional covering lattice (see Section 5.5.6).

In Section 5.6 we continue the investigation of lattices similar to E8, revealing a

previously unknown phenomenon: The existence of local lattice covering maxima,

for which we derive necessary and suﬃcient conditions. Based on them we prove

computationally that the E6 root lattice is the only lattice covering maximum in

dimensions less or equal to 6, aside of Z (which trivially is at the same time a

1-dimensional lattice covering optimum). The E8 root lattice itself turns out to be

only “almost” a local lattice covering maximum: Almost any, but not every local

change of an associated positive definite quadratic form yields a lower covering

density. We baptize these lattices covering pessima and prove that a lattice is a

covering pessimum, if the Delone polytopes attaining its covering radius are all

regular cross polytopes. This is for example the case for the root lattices D4 and

E8.

In Section 5.7 we consider local lattice covering maxima within a linear subspace

T in the space of (associated) quadratic forms. We show a connection to the famous

Minkowski conjecture, respectively to the so-called covering conjecture. Based on

our new T -secondary cone theory we obtain an algorithm which can be used to

verify or falsify the covering conjecture in a given dimension. By a recent result of

Curtis McMullen the covering conjecture implies the Minkowski conjecture. Thus