Spectrahedra and Convex Hulls of Rank-One Elements

TL;DR

This paper proves the Helton-Nie Conjecture by showing its equivalence to the statement that the convex hull of rank-one elements of a spectrahedron is a spectrahedral shadow, connecting spectrahedra with convex hulls of rank-one matrices.

Contribution

It establishes the equivalence between the Helton-Nie Conjecture and a property of spectrahedra related to convex hulls of rank-one elements, providing new insights into spectrahedral representations.

Findings

Helton-Nie Conjecture is equivalent to convex hulls of rank-one elements being spectrahedral shadows.

The equivalence holds for both general and compact convex semialgebraic sets.

Examples illustrate the relationship between spectrahedra and convex subsets of rank-one elements.

Abstract

The Helton-Nie Conjecture (HNC) is the proposition that every convex semialgebraic set is a spectrahedral shadow. Here we prove that HNC is equivalent to another propo- sition related to quadratically constrained quadratic programming. Namely, that the convex hull of the rank-one elements of any spectrahedron is a spectrahedral shadow. In the case of compact convex semialgebraic sets, the spectrahedra may be taken to be compact. We illustrate the relationship between spetrahedra and these convex subsets with examples.