Almost simplicial polytopes I. The lower and upper bound theorems

TL;DR

This paper extends classical bounds on simplicial polytopes to almost simplicial polytopes with one nonsimplex facet, providing tight bounds, characterizations, and new constructions based on a variation of the moment curve.

Contribution

It generalizes the Lower and Upper Bound Theorems to almost simplicial polytopes, introducing new constructions and characterizations for these polytopes.

Findings

Established tight lower bounds for almost simplicial polytopes.

Established tight upper bounds and characterized maximizers.

Introduced a new construction of maximizers based on a variation of the moment curve.

Abstract

We study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex facet with, say, $d+s$ vertices, called {\it almost simplicial polytopes}. We provide tight lower and upper bound theorems for these polytopes as functions of $d,n$ and $s$, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case of $s=0$. We characterize the minimizers and provide examples of maximizers, for any $d$. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest.