156 INDEX

convex

∼ body, 7

∼ cone, 3

∼ hull, 28

∼ polyhedron, 131

∼ polytope, 132

covering

∼ conjecture, 129

∼ maximum, 122

∼ optimizer, 90

∼ pessimum, 83, 126

∼ radius, 11

locally optimal lattice ∼ , 84

covering density, 11

∼ of Delone polytope, 123

Coxeter lattice, 12, 77, 108

Coxeter-Todd lattice, 45

critical value

Hermite ∼ , 24

Minkowski ∼ , 25

cross polytope, 28, 127

C-type, 75, 100

Davenport’s conjecture, 22

dead-end, 67

decomposition

Cholesky ∼ , 5

double coset ∼ , 136

Iwasawa ∼ , 4

Delone polyhedron, 56

Delone polytope, 55

∼ of quadratic form, 55

adjacent ∼ , 57

extreme ∼ , 75

GLd(Z)-equivalent ∼ , 57, 87

GLt

d

(Z)-equivalent ∼ , 57, 89

perfect ∼ , 75

T -extreme ∼ , 129

translational equivalent ∼ , 57

Delone subdivision, 57

Delone triangulation, 57

automorphism group of ∼ , 92

densest

∼ m-periodic sphere packing, 38

∼ lattice sphere packing, 8

∼ sphere packing, 9

determinant

∼ maximization problem, 85

∼ of lattice, 6

∼ surface, 29, 35

diamond property, 133

Diophantine

∼ approximation, 23

∼ equation, 1

best ∼ approximation, 23

simultaneous ∼ approximation, 24

distance of PQF, 86

domain

C-type ∼ , 75

fundamental ∼ , 17

L-type ∼ , 57

reduction ∼ , 17

Voronoi ∼ , 29

double coset, 136

dual

∼ description problem, 132

∼ polyhedral cone, 132

duality gap, 85

Eisenstein

∼ form, 52

∼ integer, 52

∼ lattice, 49

∼ perfect form, 52

ellipsoid, 3

E-perfect form, 52

equivalent

arithmetically ∼ , 2

G-∼ , 50

(G, t)-∼ , 50

T -∼ , 50, 71

translational ∼ , 57

(T, t)-∼ , 50, 71

unimodular ∼ , 77

Euclidean

∼ minimum, 111

∼ norm, 5

∼ norm best approximation, 25

∼ number field, 111

∼ unit ball, 5

eutactic

∼ quadratic form, 34

strongly ∼ , 43

extremal lattice, 34

extreme

∼ Delone polytope, 75

∼ lattice, 34

∼ quadratic form, 34

∼ ray, 132

m-∼ , 39

periodic ∼ , 43

face, 132

∼ lattice, 132

adjacent ∼ , 132

face-to-face, 36

facet, 131

lateral ∼ , 69

lower ∼ , 69

upper ∼ , 69

Fincke and Pohst algorithm, 31

fluid diamond packing, 41

fundamental domain, 17

Gaussian

∼ form, 53

∼ integer, 53