---
tag: 71VK7M83WS
kind: definition
name: Polytope combinatorics
msc: [52B05]
defines: 879SBTVE7V
uses: [0FQSDC2EQ2, 0QV2JS9RJ3]
sources:
  - url: https://en.wikipedia.org/wiki/Polyhedral_combinatorics
---

## Definition

A convex polytope is the [[convex hull|0FQSDC2EQ2]] $P$ of a finite set of points of $\R^d$.
A hyperplane $H$ supports $P$ if $P$ lies in one of the two closed half-spaces bounded by $H$.
A *face* of $P$ is $P$ itself, the empty set, or a set $P \cap H$ for a supporting hyperplane $H$.
The *face lattice* of $P$ is the set of its faces, [[partially ordered|0QV2JS9RJ3]] by inclusion.
Two polytopes are *combinatorially equivalent* if their face lattices are isomorphic as partially ordered sets.
*Polytope combinatorics* is the study of the properties of convex polytopes that depend only on their face lattice.

## Remarks

Each face is again a convex polytope, and its dimension is the dimension of its affine span; the empty face has dimension $-1$.
Faces of dimension $0$, $1$ and $\dim P - 1$ are called vertices, edges and facets.

Example: a square and a trapezium in $\R^2$ are combinatorially equivalent.
Each has $4$ vertices and $4$ edges.
Match the vertices of one with the vertices of the other in cyclic order; this matching preserves which vertices lie on which edges.

Non-example: a square and a triangle are not combinatorially equivalent, since their face lattices have $4$ and $3$ vertices.

Properties such as the number of faces of each dimension, or whether every facet is a simplex, belong to polytope combinatorics; volume and edge lengths do not.