Matrix convex sets over the Euclidean ball and polar duals of real free spectrahedra

TL;DR

This paper investigates the structure of matrix convex sets over the Euclidean ball, revealing differences between real and complex free spectrahedra and their polar duals, with implications for their extreme points and projections.

Contribution

It demonstrates that certain free spectrahedra over the Euclidean ball are not minimal matrix convex sets and explores the distinct properties of their polar duals in the real case.

Findings

The free spectrahedron from universal anticommuting unitaries is not minimal in dimension three or higher.

The polar dual of a real free spectrahedron is rarely a projection of a real free spectrahedron.

Results on spectrahedra closed under complex conjugation do not extend to free spectrahedrops.

Abstract

We show that the free spectrahedron determined by universal anticommuting self-adjoint unitaries is not equal to the minimal matrix convex set over the ball in dimension three or higher. This example, as well as other matrix convex sets over the ball, then provides context for structure results on the extreme points of coordinate projections. In particular, we show that the free polar dual of a real free spectrahedron is rarely the projection of a real free spectrahedron, contrasting a prior result of Helton, Klep, and McCullough over the complexes. We use this to show that spanning results for free spectrahedra that are closed under complex conjugation do not extend to free spectrahedrops that meet the same assumption. These results further clarify the role of the coefficient field.