QGLBT for polytopes

TL;DR

This paper extends the Generalized Lower Bound Theorem to a broader class of polytopes, providing new quantitative and topological insights, and confirms a conjecture relating to $g$-numbers for approximating smooth convex bodies.

Contribution

It generalizes the GLBT to polytopes with simplicial low-dimensional skeletons and proves a conjecture on $g$-numbers for polytopes approximating smooth convex bodies.

Findings

A quantitative version of the GLBT for general polytopes.

A topological necessary condition for vanishing toric $g_k$ entries.

Proof of Kalai's conjecture on $g$-numbers for approximating smooth convex bodies.

Abstract

We extend the assertion of the Generalized Lower Bound Theorem (GLBT) to general polytopes under the assumption that their low dimensional skeleton is simplicial, with partial results for the general case. We prove a quantitative version of the GLBT for general polytopes, and use it to give a topological necessary condition for polytopes to have vanishing toric $g_k$ entry. As another application of the QGLBT we prove a conjecture of Kalai on $g$-numbers for general polytopes approximating a smooth convex body.