On the cone of $f$-vectors of cubical polytopes

TL;DR

This paper investigates the structure of the cone formed by $f$-vectors of cubical polytopes, providing new constructions that verify part of the Cubical Generalized Lower Bound Conjecture and disproving an analogous theorem.

Contribution

It constructs specific cubical polytopes that demonstrate the nonnegative $g$-orthant is contained in the cone of $f$-vectors, confirming a part of the conjecture and showing limitations of the cubical GLLT.

Findings

The cone of $f$-vectors contains the nonnegative $g$-orthant.

A cubical analogue of the simplicial GLLT does not hold.

Constructed polytopes verify the conjecture's one direction.

Abstract

What is the minimal closed cone containing all $f$-vectors of cubical $d$-polytopes? We construct cubical polytopes showing that this cone, expressed in the cubical $g$-vector coordinates, contains the nonnegative $g$-orthant, thus verifying one direction of the Cubical Generalized Lower Bound Conjecture of Babson, Billera and Chan. Our polytopes also show that a natural cubical analogue of the simplicial Generalized Lower Bound Theorem does not hold.