Polyhedral Subdivisions and Projections of Polytopes Jörg Rambau Dissertation (Advisor: Günter M. Ziegler, TU Berlin)

Index

$1$-connected : 10.2

$A$-discriminant : 7.2.5

$A$-resultant : 7.2.5

abstract simplicial complex : 10.4 | 10.4

admissible

ordering : 9.7

pairs of simplices : 9.2

alternating oriented matroid : 7.2.3

anti-isomorphism of posets : 10.1

antistar : 10.4

arrangement of hyperplanes : 10.5

associahedron : 7.1.2

atom : 10.1

atomic : 10.1

barycenter : 8.3

basis : 10.2

basket ball obstruction : 8.4 to 8.4 | 8.5 | 8.5

Baues conjecture : 7.2.4

big face lattice : 10.5

bipyramid : 10.3

bistellar

equivalence : 7.1.1

operation : 7 | 7.1.1 | 7.1.2 | 7.2.1 | 7.2.1 | 7.2.1 | 7.2.1 | 7.2.1 | 9 | 9.1

Bohne-Dress theorem : 7.2.3 | 7.2.3

boundary

of a cone : 10.3

of a polytope : 10.3

of an inversion set : 9.7

bounded poset : 10.1

cellular

string : 7.2.4

string complex : 7.2.4

central set

of a simplex : 9.8

of an inversion set : 9.8

chain in a poset : 10.1

chamber : 7 | 7.1.2 | 8.2 | 8.2 | 8.2 to 8.2 | 8.4

chamber complex : 7.1.2 | 8.2 | 8.2 | 8.2 to 8.2 | 8.4 | 8.5 | 8.5

characteristic

function : 7.2.5

section : 9.3 | 9.5 | 9.5

circuit : 9.2

closed

normal cone : 10.3

path : 10.2

coatom : 10.1

coatomic : 10.1

cocircuit : 10.5

coherent subdivision : 7.1.2 | 7.1.2 | 7.1.2 | 7.2.5 | 8.1

coloop : 10.5

combinatorial

cyclic polytope : 9.3

differential manifold : 7.2.3

facet : 9.2

join : 9.2

link : 9.2

model : 9.2

for a loop space : 7.2.4

for the Grassmannian : 7.2.4

polytope : 9.2

simplex : 9.2

simplicial complex : 9.2

subcomplex : 9.2

triangulation : 9.2

combinatorially isomorphic

complexes : 10.4

subdivisions : 7.1.1

common refinement

of fans : 10.3

of subdivisions : 7.1.1

complete fan : 10.3

composition : 10.5

compression in a triangulation : 9.4

concatenation of paths : 8.3 | 10.2

conical hull : 10.3

consistent : 9.7

continuous : 10.2

at a point : 10.2

contractible : 10.2

contraction of an element

in a set : 9.1

in a triangulation : 9.4

in an inversion set : 9.7

in an oriented matroid : 10.5

covector : 10.5

cover in a poset : 10.1

covering of a topological space : 10.2

covering relation : 10.1

Coxeter groups : 7.1.2

cross-polytope : 10.3

cyclic polytope : 7 | 7 | 7 | 7.1.1 | 7.1.1 | 7.1.1 | 7.1.1 | 7.1.1 | 7.1.1 | 7.1.2 | 7.1.2 | 7.2.2 | 9.1 | 9.1 | 9.3 to 9.3 | 10.3 | 10.3

degenerate : 8.1

Delaunay triangulation : 7.2.1

deletion of an element

in a set : 9.1

in a triangulation : 9.4

in an inversion set : 9.7

in an oriented matroid : 10.5

diameter : 8.3 | 10.2

dimension

of a cone : 10.3

of a polytope : 10.3

of an abstract simplicial complex : 10.4

direct map : 9.8

direct sum : 10.3

double description : 10.3

double stippling : 7.2.4

edge : 10.3

edge-flipping

diameter : 7.2.1

distance : 7.2.1

operation : 7.2.1

equivalent

admissible orderings : 9.7

locally coherent functions : 8.2

even gap : 9.3 | 9.5

expansion

in a triangulation : 9.4

of an inversion set : 9.7

extension : 10.5

of a triangulation : 9.4

of an inversion set : 9.7

poset : 7.2.3

space : 7.2.3

extension space conjecture : 7.2.3

face

of a combinatorial simplex : 9.2

of a cone : 10.3

in direction $\psi$ : 10.3

of a polytope : 10.3

in direction $\psi$ : 8.2 | 10.3

face lattice

of a cone : 10.3

of a polytope : 8.2 | 10.3

facet

of a combinatorial simplex : 9.2

of a complex : 10.4

of a cone : 10.3

of a polytope : 9.2 | 10.3

fan : 10.3

fiber : 7 | 7.1.2 | 8.2 | 8.2 | 8.2 to 8.2

fiber polytope : 7.1.2 | 8.1

flip map : 9.8

free

lower facet : 9.6

oriented matroid : 10.5

upper facet : 9.6

full-dimensional

cone : 10.3

polytope : 10.3

functorial

circuit properties : 9.3

circuit-facet relations : 9.3

facet properties : 9.3

flip properties : 9.5

order properties : 9.5

relations : 9.8

Gale's evenness criterion : 9.3

generalized Baues conjecture : 7 | 7 | 7 | 7.1.2 | 7.1.2 | 7.1.2 | 8.1 | 8.4 to 8.4

generalized bistellar operation : 7.1.2

generator : 10.2

geometric

interpretation : 9.2 | 9.5 | 9.5

simplex : 9.2

simplicial complex : 9.2

subcomplex : 9.2

globally coherent string : 8.1 | 8.3

graded poset : 10.1

graph of all triangulations : 7.1.1

Grassmannian : 7.2.4

greedily triangulable : 9.4

Hasse diagram : 10.1

higher Bruhat order : 7.2.2 | 9.1 | 9.1 | 9.7 | 9.8 to 9.8

higher Stasheff-Tamari order

first : 7.2.2 | 9.1 | 9.1 | 9.5

second : 9.1 | 9.5

homeomorphism : 10.2

homotopic : 10.2

relative $\partial [0, 1]$ : 8.3 | 10.2

homotopy

equivalence : 10.2

equivalent : 10.2

of functions : 10.2

relative a subset : 10.2

hypercube : 10.3

increasing

bistellar flip set : 9.5

bistellar operation : 7.2.2 | 9.1 | 9.5

flip function : 9.5

flip set : 9.5

incremental triangulation : 7.2.1

index : 9.8

induced subdivision : 7.1.2 | 7.1.2 | 7.1.2 | 8.1

intersection property : 9.2

interval in a poset : 10.1

inversion set : 9.7

isomorphism of posets : 10.1

iterated fiber polytope : 7.1.2

iterated loop space : 7.2.4

join

in a poset : 10.1

of polytopes : 10.3

of simplicial complexes : 10.4

$k$-neighborly : 10.3 | 10.3

$k$-packet : 9.7

label set : 9.2

labelling function : 9.2

lattice : 10.1

left boundary

function : 9.8

of a simplex : 9.8

length of a chain : 10.1

lexicographic order : 9.7

lifting : 10.5

of a path : 8.3 | 10.2

lifts weakly lower : 9.5

link : 8.5 | 10.4

locally coherent

function : 8.2 | 8.3 | 8.3 | 8.4 to 8.4

function values : 8.3 | 8.5

string : 7.1.2 | 8.1 | 8.2 | 8.3 | 8.3 | 8.4 to 8.4 | 8.5 | 8.5 | 8.5

loop : 10.5

lower

face : 7.1.1 | 7.1.1

facet : 9.1 | 9.3 | 9.5 | 9.6 to 9.6

lower bound in a poset : 10.1

MacPhersonian : 7.2.4

maximal element in a poset : 10.1

meet in a poset : 10.1

metric space : 10.2

minimal

combinatorial dependence : 9.2

element in a poset : 10.1

Minkowski sum : 10.3

model theorem for loop spaces : 7.2.4

monotone path polytope : 7.1.2

negative part

of a circuit : 9.2

of a sign vector : 10.5

neighborhood : 10.2

neighborly : 10.3

Newton polytope : 7.2.5

non-degenerate : 8.1

normal cone : 8.2 | 10.3

over a chamber : 8.2 | 8.2 to 8.2 | 8.4 to 8.4

normal fan : 10.3

over a chamber : 8.2 | 8.2 to 8.2 | 8.4 to 8.4 | 8.5

normally equivalent : 8.2

null-homotopic : 10.2

odd gap : 9.3 | 9.5

OM-Grassmannian : 7.2.4

one-element lifting : 10.5

open

ball : 10.2

set : 10.2

order complex : 7.1.2 | 7.2.3 | 7.2.4 | 7.2.4 | 8.1 | 10.1 | 10.1

order-preserving map : 10.1

order-reversing map : 10.1

oriented matroid : 10.5

of a vector configuration : 10.5

of a zonotope : 7.2.3

parametrized cyclic polytope : 9.3

partial order : 10.1

partial triangulation : 7.2.1 | 9.4

partially ordered set : 10.1

path : 10.2

path space : 7.2.4

path-connected : 10.2

permutahedron : 7.2.4

permuto-associahedron : 7.1.2

piecewise linear section : 7.1.2

planar bistellar operation : 7.1.1

pointed

cone : 10.3

fan : 10.3

polyhedral

cone : 10.3

subdivision : 10.4

of a point configuration : 7.1.1 | 7.1.2

of a polytope : 7.1.1

polytopal complex : 10.4

polytope : 9.2 | 10.3

polytope projection : 7 | 7.1.2 | 8.1 | 8.2 to 8.2 | 8.4 to 8.4 | 8.5 | 8.5 | 9.1 | 9.3

poset : 10.1

positive part

of a circuit : 9.2

of a sign vector : 10.5

principal $A$-determinant : 7.2.5

prism : 10.3

product

of metric spaces : 8.3 | 10.2

of polytopes : 10.3

proper

face

of a cone : 10.3

of a polytope : 10.3

part of a poset : 10.1

pseudosphere arrangement : 10.5

pure : 10.4

pyramid : 10.3

rank

in a poset : 10.1

of a poset : 10.1

of an oriented matroid : 10.5

ranked poset : 10.1

ray : 10.3

real Grassmann manifold : 7.2.4

realizable oriented matroid : 7.2.4 | 10.5

refinement

of a fan : 10.3

of a subdivision : 7.1.1

reflex vertex : 7.2.1

regular

subdivision : 7.1.1 | 7.1.2

triangulation : 7.1.1 | 7.1.2 | 9.1

triangulation for weights : 7.2.1

regular triangualtion : 7.1.1

relative interior

of a cone : 10.3

of a polytope : 10.3

restricted polyhedral subdivision : 7.1.2 | 9.1

restriction : 10.5

right boundary

function : 9.8

of a simplex : 9.8

rotations in binary trees : 7.2.2

secondary polytope : 7.1.2 | 7.1.2 | 9.1

example : 7.2.5

via characteristic functions : 7.2.5

via fiber polytope : 7.1.2

separation : 10.5

shellability : 10.3

shellable : 10.4

shelling : 10.4

sign vector : 10.5

simple polygon : 7.2.1

simplex : 10.3

simplicial

complex : 10.4

polytope : 9.2

single-element extension : 10.5

spherical desuspension : 7.2.4

standard

cyclic polytope : 10.3

star : 10.4

strong image : 10.5

strongly

euclidean oriented matroid : 7.2.3

non-degenerate : 8.1

subbasis : 10.2

support

of a bistellar operation : 7.1.1

of a circuit : 9.2

of a sign vector : 10.5

of a simplicial complex : 10.4

supporting hyperplane

of a cone : 10.3

of a polytope : 10.3

Tamari lattice : 7.2.2

tight

induced subdivision : 7.1.2

locally coherent string : 8.1 | 8.4 | 8.4 | 8.5 | 8.5

topological space : 10.2

topology : 10.2

induced by a metric : 10.2

transitive closure : 10.1

triangulation : 7

of a cyclic polytope : 7 | 7.1.1 | 7.1.2 | 7.2.2 | 9.1 | 9.4 to 9.4 | 9.5 to 9.5 | 9.6 to 9.6 | 9.8 | 9.8

of a point configuration : 7.1.1 | 7.1.2 | 7.2.1 to 7.2.1 | 7.2.2 | 7.2.3 | 7.2.5 | 8.1 | 9.1 | 9.2

of a polytope : 7.1.1

of a topological space : 7.2.4 | 8.1

of an oriented matroid : 7.2.3

trivial face of a polytope : 8.2

twist function : 8.3

twisted capped prism : 9.5

underlying set of a complex : 10.4

union property : 9.2

universal covering : 8.3 | 10.2

upper bound in a poset : 10.1

upper bound theorem : 10.3 | 10.3 | 10.4

upper facet : 9.1 | 9.3 | 9.6 to 9.6

vertex : 10.3

weak Bruhat order : 9.1 | 10.1

weak image : 10.5

weak map relation : 10.5

weakly

neighborly polytope : 7.1.1

non-degenerate : 8.1

zonotopal tiling : 7.2.3

zonotope : 7.2.3