Edges of the Barvinok-Novik orbitope

TL;DR

This paper proves a conjecture about the exact threshold for edges in the Barvinok-Novik orbitope, confirming the tightness of the lower bound for all symmetric trigonometric moment curves.

Contribution

It establishes the conjecture that the lower bound for the edge threshold in the Barvinok-Novik orbitope is tight for all values of k, extending previous results.

Findings

Confirmed the tightness of the threshold bound for all k

Extended the result of Smilansky for k=2 to all k

Provided a complete characterization of edges in the orbitope

Abstract

Here we study the k^th symmetric trigonometric moment curve and its convex hull, the Barvinok-Novik orbitope. In 2008, Barvinok and Novik introduce these objects and show that there is some threshold so that for two points on S^1 with arclength below this threshold, the line segment between their lifts on the curve form an edge on the Barvinok-Novik orbitope and for points with arclenth above this threshold, their lifts do not form an edge. They also give a lower bound for this threshold and conjecture that this bound is tight. Results of Smilansky prove tightness for k=2. Here we prove this conjecture for all k.