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Hypergraph polytopes. (English) Zbl 1222.05193
Summary: We investigate a family of polytopes introduced by E.M. Feichtner, A. Postnikov and B. Sturmfels, which were named nestohedra. The vertices of these polytopes may intuitively be understood as constructions of hypergraphs. Limit cases in this family of polytopes are, on the one end, simplices, and, on the other end, permutohedra. In between, as notable members one finds associahedra and cyclohedra. The polytopes in this family are investigated here both as abstract polytopes and as realized in Euclidean spaces of all finite dimensions.
The later realizations are inspired by J.D. Stasheff’s and S. Shnider’s realizations of associahedra. In these realizations, passing from simplices to permutohedra, via associahedra, cyclohedra and other interesting polytopes, involves truncating vertices, edges and other faces. The results presented here reformulate, systematize and extend previously obtained results, and in particular those concerning polytopes based on constructions of graphs, which were introduced by M. Carr and S.L. Devadoss.

##### MSC:
 05C65 Hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory 52B11 $$n$$-dimensional polytopes 51M20 Polyhedra and polytopes; regular figures, division of spaces 55U05 Abstract complexes in algebraic topology 52B12 Special polytopes (linear programming, centrally symmetric, etc.)
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