Hypergraph Polytopes

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.