Hypergraph Polytopes

TL;DR

This paper explores a family of polytopes called nestohedra, which are constructed from hypergraphs, unifying various well-known polytopes like simplices, permutohedra, associahedra, and cyclohedra, through systematic geometric realizations.

Contribution

It reformulates, extends, and systematizes the theory of hypergraph-based polytopes, connecting them with known polytopes and providing new geometric realizations.

Findings

Unified framework for nestohedra and related polytopes

Realizations inspired by associahedra constructions

Extension of previous results on graph-based polytopes

Abstract

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.