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Schlegel Diagrams 31 We are interested in polyhedral complexes since a polytope P gives rise to some natural polyhedral complexes which play an important role in the study of polytopes. 17. Let P be a polytope. (1) The complex C(P ) of the polytope P is the polyhedral complex of all faces of P . The face poset of C(P ) is the face lattice of P . (2) The boundary complex ∂(C(P )) is the polyhedral complex of all the proper faces of P along with the empty face. (3) A polytopal subdivision of P is a polyhedral complex C with support P in which all the polyhedra are polytopes.

The cubes Cd are simple polytopes while the crosspolytopes Cd∆ are simplicial polytopes. 15. Can you tell from the face lattice of a polytope whether the polytope is simple or simplicial? 16. (1) Show that a polygon (a 2-polytope) is both simple and simplicial. (2) Construct a polytope that is neither simple nor simplicial. 17. If P ⊂ Rd is a d-polytope with the origin in its interior, then the polar of P is the d-polytope P ∆ := {y ∈ Rd : y · x ≤ 1 for all x ∈ P }. 3. Faces of Polytopes 21 Check that cubes are polar to cross polytopes.

Wp } ⊂ Rq be a vector configuration. 6. Bizarre Polytopes 49 (1) A circuit of W is any non-zero vector u ∈ Rp of minimal support such that w1 u1 +. +wp up = 0. The vector sign(u) is called a signed circuit of W. (2) A co-circuit of W is any non-zero vector of minimal support of the form (v·w1 , . . , v·wn ) where v ∈ Rq . The sign vector of a co-circuit is called a signed co-circuit. 5. Consider the vector configuration shown in Figure 1 that is the Gale transform of the triangular prism from Chapter 5.