The boundary complex of a convex polytope 
is the collection of its nonempty proper
faces, together with their containment relations. Here a
proper face is any face other than
itself. Some conventions also include
the empty set as a face. The
union of the faces is the topological boundary
of 
,
but the complex also records how that boundary is divided
into faces (Ziegler 1995).
Every nonempty face of a member belongs to the complex, and two members intersect in a common face or not at all. If all its members are simplices, the boundary complex is a simplicial complex. For example, the boundary complex of a tetrahedron has 4 vertices, 6 edges, and 4 triangular faces. The boundary complex of a cube has 8 vertices, 12 edges, and 6 square faces, so it is not a simplicial complex. Subdividing each square face into two triangles gives a simplicial triangulation of the same boundary.
The boundary of a 
-dimensional convex polytope,
for 
,
is homeomorphic to 
. Thus a simplicial boundary complex of a three-dimensional
convex polytope triangulates a sphere.
The term is also used for the face decomposition of the surface of a nonconvex polyhedron. In this sense, the boundary complex of the Császár polyhedron is the simplicial complex formed by its 7 vertices, 21 edges, and 14 triangular faces. It triangulates a torus and cannot be the boundary complex of a convex polytope (Kühnel and Lutz 2000). This surface complex is not the intrinsic boundary of the triangulated torus, which is a closed surface and has no manifold boundary.
