Math 2.0

Convex and discrete geometry Definition

Polytope combinatorics

Builds on Convex hullPartial order

Definition

A convex polytope is the convex hull PP of a finite set of points of ℝd\R^d. A hyperplane HH supports PP if PP lies in one of the two closed half-spaces bounded by HH. A face of PP is PP itself, the empty set, or a set P∩HP \cap H for a supporting hyperplane HH. The face lattice of PP is the set of its faces, partially ordered 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 −1-1. Faces of dimension 00, 11 and dim⁡P−1\dim P - 1 are called vertices, edges and facets.

Example: a square and a trapezium in ℝ2\R^2 are combinatorially equivalent. Each has 44 vertices and 44 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 44 and 33 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.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Convex hull, concept, no evidence levelConvex hullconceptPartial order, concept, no evidence levelPartial orderconceptPolytope combinatorics, definition, no evidence levelPolytope combinatoricsdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Convex hull concept
  2. Partial order concept
  3. Polytope combinatorics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Polyhedral_combinatorics
About this page
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URIm2://71VK7M83WS
Kinddefinition
WritingW0 stub
MSC52B05
DefinesPolytope combinatorics
UsesConvex hull, Partial order
Changed2 November 2026

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