Axioms for the g-vector of general convex polytopes

TL;DR

This paper proposes axioms to extend McMullen's g-vector from simple to general convex polytopes, supported by computational evidence and leading to new inequalities on polytope invariants.

Contribution

It introduces axioms for the g-vector extension to all convex polytopes and conjectures a finite computational method for its formula in each dimension.

Findings

Computer calculations for 5-dimensional polytopes support the conjecture.

New linear inequalities on convex polytope flag vectors are derived.

A hypothesized higher-order homology explains the extension of the g-vector.

Abstract

McMullen's g-vector is important for simple convex polytopes. This paper postulates axioms for its extension to general convex polytopes. It also conjectures that, for each dimension d, a stated finite calculation gives the formula for the extended g-vector. This calculation is done by computer for d=5 and the results analysed. The conjectures imply new linear inequalities on convex polytope flag vectors. Underlying the axioms is a hypothesised higher-order homology extension to middle perversity intersection homology (order-zero homology), which measures the failure of lower-order homology to have a ring structure.