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, [9] 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 F.

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, Adv.

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.