Local neighborliness of the symmetric moment curve

Abstract

A centrally symmetric analogue of the cyclic polytope, the bicyclic polytope, was defined in [BN08]. The bicyclic polytope is defined by the convex hull of finitely many points on the symmetric moment curve where the set of points has a symmetry about the origin. In this paper, we study the Barvinok-Novik orbitope, the convex hull of the symmetric moment curve. It was proven in [BN08] that the orbitope is locally $k$-neighborly, that is, the convex hull of any set of $k$ distinct points on an arc of length not exceeding ${\varphi }_k$ in ${\mathbb{S}}^1$ is a $(k-1)$-dimensional face of the orbitope for some positive constant ${\varphi }_k$. We prove that we can choose ${\varphi }_k$ bigger than $\gamma k^{-3/2}$ for some positive constant $\gamma$.