Neighborliness of the symmetric moment curve

TL;DR

This paper studies the convex hull of the symmetric moment curve in high-dimensional space, establishing the maximum arc length for local neighborliness and constructing polytopes with many faces.

Contribution

It characterizes the maximum neighborliness arc length for the symmetric moment curve and constructs highly faceted symmetric polytopes.

Findings

Maximum arc length phi_k exceeds pi/2 for all k

Limit of phi_k approaches pi/2 as k increases

Constructs centrally symmetric polytopes with record number of faces

Abstract

We consider the convex hull B_k of the symmetric moment curve U(t)=(cos t, sin t, cos 3t, sin 3t, ..., cos (2k-1)t, sin (2k-1)t) in R^{2k}, where t ranges over the unit circle S= R/2pi Z. The curve U(t) is locally neighborly: as long as t_1, ..., t_k lie in an open arc of S of a certain length phi_k>0, the convex hull of the points U(t_1), ..., U(t_k) is a face of B_k. We characterize the maximum possible length phi_k, proving, in particular, that phi_k > pi/2 for all k and that the limit of phi_k is pi/2 as k grows. This allows us to construct centrally symmetric polytopes with a record number of faces.