A polyhedral complex is a collection of convex polyhedra (of various dimensions) in a real vector space glued together along common faces so that the intersection of any two polyhedra is a face of each, producing a cell-like stratified structure.

Principle

The organizing rule is face-to-face compatibility: cells are convex polyhedra, their faces are included as lower-dimensional cells, and the intersection condition enforces a combinatorial and geometric coherence across the complex.

Demonstration

Example: a simplicial complex is a special case where all polyhedra are simplices; more generally Voronoi decompositions, polyhedral fans, and cell decompositions of polyhedral manifolds are instances of polyhedral complexes.

Misapplication

Treating an arbitrary cell decomposition whose cells are not convex polyhedra or where intersections are not faces (for example overlapping cells with nonface intersection) as a polyhedral complex misapplies the definition.

Consequence

Correct recognition allows combinatorial invariants (f-vector, face poset), piecewise-linear methods, and computations of topological invariants via CW-type arguments and lends itself to polyhedral and tropical geometry techniques.

Reversal

The reversal is a general CW-complex or stratified space without convexity constraints: cells may be arbitrary topological disks and intersections need not be faces of convex polyhedra, losing linear structure.

Boundary

Scope: finite or locally finite collections of convex polyhedra in real vector spaces with face-to-face gluing; excludes nonconvex cells, complexes defined in nonreal or nonvector settings without an affine structure, and decompositions failing the face intersection property.

Semantic Tension

Tension occurs between 'polyhedral complex' and 'simplicial complex' or 'CW-complex': simplicial complexes impose simplex structure and rigidity, while CW-complexes are more flexible but drop polyhedral convexity and linearity.

Synthesis

A polyhedral complex is a face-to-face assembly of convex polyhedra forming a stratified, cell-like structure: convexity and face-intersection rules give it linear and combinatorial control, enabling piecewise-linear topology and polyhedral-algebraic methods.