On stretching the interval simplex-permutohedron

TL;DR

This paper explores extending the family of nestohedra polytopes by modifying their interval boundaries using an iterative nested set complex construction, simplifying the algebraic representation.

Contribution

It introduces a method to extend nestohedra intervals through iterative nested set complexes with simplified algebraic descriptions.

Findings

Extended nestohedra intervals via iterative nested set complexes

Simplified algebraic representation of nested sets

Avoided increased complexity in simplicial complex structures

Abstract

A family of polytopes introduced by E.M. Feichtner, A. Postnikov and B. Sturmfels, which were named nestohedra, consists in each dimension of an interval of polytopes starting with a simplex and ending with a permutohedron. This paper investigates a problem of changing and extending the boundaries of these intervals. An iterative application of Feichtner-Kozlov procedure of forming complexes of nested sets is a solution of this problem. By using a simple algebraic presentation of members of nested sets it is possible to avoid the problem of increasing the complexity of the structure of nested curly braces in elements of the produced simplicial complexes.