A lower bound theorem for centrally symmetric simplicial polytopes

Abstract

Stanley proved that for any centrally symmetric simplicial $d$-polytope $P$ with $d\ge 3$, $g_2(P)\ge \left(\genfrac{}{}{0ex}{}{d}{2}\right)-d$. We provide a characterization of centrally symmetric $d$-polytopes with $d\ge 4$ 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.