158 INDEX

Minkowski

∼ conjecture, 128

∼ critical value, 25

∼ geodesic MCF, 25

∼ reduction, 19

∼ ’s algorithm, 20

∼ ’s second fundamental theorem, 22

lexicographical ∼ reduced, 25

Minkowski-Hlawka, 9

moduli space, 37

moment of inertia, 97

Mordell type inequality, 54

Morse function, 128

m-perfect, 40

m-periodic (point) set, 37

norm, 5

normal cone, 29

normalizer, 50

number field

Euclidean ∼ , 111

thin ∼ , 111

totally real ∼ , 110

weakly thin ∼ , 111

open

∼ polyhedral subdivision, 51

∼ ray, 3

oriented matroid, 62

orthogonal complement, 42

outer

∼ coeﬃcient, 4

∼ normal cone, 94

packing radius, 6, 9

packing-covering

∼ constant, 13

∼ optimizer, 90

∼ problem, 13

locally optimal lattice ∼ , 84

parallelotope, 5

partial quotient matrices, 24

partition backtrack, 136

patchwork lattice, 49

perfect

strongly ∼ , 44

perfect Delone polytope, 75

perfect form, 29

contiguous ∼ , 30

Eisenstein ∼ , 52

Gaussian ∼ , 53

Hurwitz quaternionic ∼ , 54

Voronoi’s first ∼ , 31

periodic extreme set, 43

periodic form, 37

periodic set, 10, 27

m-∼ , 37

rational ∼ , 46

standard ∼ , 37

pessimum, 12, 126

point density

upper, lower ∼ , 9

polyhedral

∼ complex, 75, 133

∼ subdivision, 35

polyhedral cone, 132

automorphism group of ∼ , 74

GLt

d

(Z)-equivalent ∼ , 74

rational normalized ∼ , 73

polyhedron, 131

locally finite ∼ , 28

vertex-set of ∼ , 60

polytope, 132

Gosset ∼ , 75, 126

regular cross ∼ , 127

repartitioning ∼ , 69

Schl¨ afli ∼ , 75, 126

secondary ∼ , 57

positive

∼ definite quadratic form, 2

∼ semidefinite quadratic form, 4

PQF (see also under quadratic form), 2

principle congruence subgroup, 46

projective automorphism group, 134

quadratic form, 1

∼ rigid, 114

adjoint ∼ , 35

characteristic ∼ , 73

Eisenstein ∼ , 52

E-perfect ∼ , 52

eutactic ∼ , 34

extreme ∼ , 34

Gaussian ∼ , 53

GLt

d

(Z)-equivalent ∼ , 48

G-perfect ∼ , 49

G-perfect ∼ , 53

(G, t)-perfect ∼ , 49

H-perfect ∼ , 54

Hurwitz quaternionic ∼ , 53

Minkowski reduced ∼ , 19

perfect ∼ , 29

positive definite ∼ , 2

positive semidefinite ∼ , 4

rank-1 ∼ , 29

rational, real, integral ∼ , 1

reduced ∼ , 17

strongly eutactic ∼ , 43

strongly perfect ∼ , 44

t-eutactic ∼ , 47

t-extreme ∼ , 47

T -perfect ∼ , 49

t-perfect ∼ , 47

(T, t)-perfect ∼ , 49

quantizer problem, xv

randomized algorithm, 72

rank-1 form, 29