ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

13. Combinatorics

1991 MS Classification: 52,5 Santos, Francisco, Universidad de Cantabria, Spain Applications of the Cayley trick to triangulations of polytopes. The polyhedral Cayley trick gives a bijection between mixed polyhedral subdivisions of the (d-dimensional) Minkowski sum of n d-polytopes and all polyhedral subdivisions of their Cayley embedding (which is (n + d − 1)- dimensional). We use this to study subdivisions of high dimensional polytopes which can be expressed as Cayley embeddings of smaller dimensional ones:

• This gives a new proof of the Bohne-Dress theorem on zonotopal tilings based on regarding Lawrence polytopes as Cayley embeddings of segments.

• Δ_{k} × Δ_{2} is the Cayley embedding of k + 1 equal triangles. Its triangulations thus correspond to lozenge tilings of a triangle of side k + 1. This allows enumeration, visualization and proving flip-connectivity of such triangulations.

• We construct triangulations of d-cubes with assymptotically d!ρ^{d} simplices, for ρ = .836. This improves on the previous smallest value of ρ and involves much simpler computations.