Convex polytopes, faces, and f-vectors

A convex polytope is the convex hull of a nonempty finite set of points in Euclidean space. Equivalently, it is a nonempty bounded intersection of finitely many closed halfspaces. Its dimension is the dimension of its affine hull, which can be smaller than the ambient coordinate dimension. A triangle sitting in three-dimensional space is still a two-dimensional polytope. Work inside the affine hull when discussing facets or full-dimensional supporting inequalities.

For a nonzero linear functional on that affine hull, the points where its maximum over the polytope is attained form an exposed face. Every nonempty proper face of a polytope can be obtained this way. Vertices have dimension zero, edges dimension one, and facets dimension d − 1 for a d-dimensional polytope. It is often convenient to include the empty face and the whole polytope in the face lattice, but we will exclude both from the ordinary f-vector.

Write fᵢ for the number of i-dimensional proper faces, with indices from zero through d − 1. Thus the ordinary f-vector of a three-polytope is its numbers of vertices, edges, and facets. Euler’s relation is an alternating sum, not a formula for each entry independently. In dimension three it becomes vertices minus edges plus facets equals two, because the boundary is a cell decomposition of a sphere.

Double counting supplies further constraints in dimension three. Each edge meets two facets, and each facet has at least three edges, so three times the facet count is at most twice the edge count. Similarly every vertex has at least three incident edges. These inequalities distinguish plausible counts from impossible ones, but they do not give the full incidence graph. Different higher-dimensional polytopes can share the same f-vector, and even identical metric volume reveals little about these counts.

A d-polytope is simple if each vertex has exactly d incident edges; it is simplicial if every facet is a simplex. The cube is simple, while the octahedron is simplicial. A simplex has both properties. In dimension three, equality in the vertex-incidence bound characterizes simplicity, while equality in the facet-incidence bound characterizes triangular facets. Confusing these two conditions reverses the roles of vertices and facets in later duality arguments.

An invertible affine map preserves convex combinations, dimensions, and supporting faces, so it preserves the entire face lattice and hence the f-vector. It need not preserve angles, lengths, or Euclidean volume. Projection is different: a noninvertible linear map can merge vertices or collapse edges. Before transferring a combinatorial conclusion from a transformed picture, check whether the map is actually invertible on the polytope’s affine hull. This check separates harmless coordinate changes from operations that change the object being counted.