# A lower bound theorem for centrally symmetric simplicial polytopes

## TL;DR

This paper characterizes centrally symmetric simplicial polytopes that meet the lower bound on their g_2 invariant, extending the classical Lower Bound Theorem to symmetric cases.

## Contribution

It provides a complete characterization of equality cases in the lower bound for centrally symmetric simplicial polytopes, generalizing a fundamental theorem.

## Key findings

- Characterization of polytopes satisfying the equality in the lower bound
- Extension of the classical Lower Bound Theorem to symmetric polytopes
- Identification of structural properties of extremal polytopes

## Abstract

Stanley proved that for any centrally symmetric simplicial $d$-polytope $P$ with $d\geq 3$, $g_2(P) \geq {d \choose 2}-d$. We provide a characterization of centrally symmetric $d$-polytopes with $d\geq 4$ that satisfy this inequality as equality. This gives a natural generalization of the classical Lower Bound Theorem for simplicial polytopes to the setting of centrally symmetric simplicial polytopes.