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Cell decomposition of polytopes by bending. (English) Zbl 0673.52005
Let P be a convex d-polytope and H a hyperplane meeting P but not any vertex of P. It is shown that a cell decomposition of P exists consisting of the convex hulls of pairs of faces of P, such that the faces in each pair are separated by H and their dimensions sum to d-1.
To prove this result the authors introduce the idea of “bending a polytope around hyperplane”, yielding a \(d+1\)-polytope, which, when flattened back into d-space, gives rise to the sought cell decomposition.
A similar technique shows that the region between two convex d-polytopes (either disjoint or one contained in the other) may be likewise cell- decomposed by using pairs of faces, one of each polytope.
Reviewer: F.Plastria

MSC:
52Bxx Polytopes and polyhedra
51M20 Polyhedra and polytopes; regular figures, division of spaces
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