Local neighborliness of the symmetric moment curve

TL;DR

This paper investigates the local neighborliness properties of the Barvinok-Novik orbitope, showing that the size of arcs guaranteeing face structure can be bounded below by a function decreasing as a power of k.

Contribution

The authors establish a lower bound on the arc length parameter _k for the Barvinok-Novik orbitope, improving understanding of its local neighborliness.

Findings

The bound _k > b3 k^{-3/2} is proven for some positive b3.

The result extends previous work by providing explicit bounds on _k.

The orbitope exhibits local neighborliness properties with quantifiable arc length bounds.

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 $\phi_k$ in $\mathbb{S}^1$ is a $(k-1)$-dimensional face of the orbitope for some positive constant $\phi_k$. We prove that we can choose $\phi_k $ bigger than $\gamma k^{-3/2} $ for some positive constant $\gamma$.