Note that any convex polytope
can be written in the form [[PI].
5] shows that every arrangement of spheres (and hence every central arrangemen of hyperplanes) is combinatorially equivalent to some convex polytope
,  proved that there is a relation between the number of lattice point on a sphere and the volume of it.
j] (F) denote the set of all j-dimensional faces of a convex polytope
The number of faces of a simplicial convex polytope
A d-dimensional associahedron is a simple convex polytope
whose vertices correspond to the triangulations of a convex (d + 3)-gon and whose edges correspond to flips between these triangulations.
It is also equivalent to the existence of an algorithm that decides if a given lattice is isomorphic to the face lattice of a convex polytope
in rational Euclidean space.
We call a triangulation of a convex polytope
unimodular if every simplex in the triangulation has normalized volume one.
The faces of a convex polytope
and the cells of a regular cell complex are examples of Whitney stratifications, but in general, a stratum in a stratified space need not be contractible.
Stanley, The number of faces of a simplicial convex polytope
A lattice polytope P is a convex polytope
whose vertices lie in a lattice N contained in the vector space [R.
Then the associahedron is the convex polytope
in which each vertex corresponds to a way of correctly inserting the parentheses, and the edges correspond to single application of the associativity rule.
A tessellation of the 3-dimensional Euclidean space is a countable and locally finite family of convex polytopes
, the cells of the tessellation.