By Rayward-Smith V.I. (ed.)
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Extra info for Combinatorial Optimization 2
Poset of D along with the faces F and P added in. How does F sit in this poset? For a face G in D, G ⊆ F if and only if G is a face of F . The Schlegel diagram thus completely encodes the combinatorics of an d-dimensional polytope into a (d − 1)-dimensional complex. This is especially useful for d ≤ 4. 21. What polytope has the Schlegel diagram shown in Figure 8? Draw the face lattice of this polytope. Does this polytope have a different Schlegel diagram if you look through a different facet? Figure 8.
12. [Stu88] There is a 4-polytope P with 7 facets for which the shape of a facet cannot be prescribed. Proof. Let P ∆ be the bi-pyramid over matrix for this be: 1 1 1 1 1 0 −1 0 A= 0 −1 0 1 0 0 0 0 0 0 0 0 a square pyramid. Let the A 1 0 0 1 0 1 1 0 0 0 0 1 0 1 −1 . To see that the convex hull of the columns of A is a bi-pyramid over a square pyramid, first note that the convex hull of the first four columns of A is a square, and the convex hull of the first five columns of A is a square pyramid.
2) (P ∆ )∆ = P . (3) The polars of simple polytopes are simplicial and the polars of simplicial polytopes are simple. (4) The face lattices F(P ) and F(P ∆ ) are anti-isomorphic. This means that there is a bijection between the k-faces of P and the (d − k − 1)-faces of P ∆ , where we assume that dim(P ) = dim(P ∆ ) = d, that also “inverts” the inclusions. Informally, the Hasse diagram of (F(P ), ⊆) can be gotten by rotating the Hasse diagram of (F(P ∆ ), ⊆) by 180 degrees. In the rest of this chapter, we focus on a special class of polytopes called cyclic polytopes.
Combinatorial Optimization 2 by Rayward-Smith V.I. (ed.)