Extreme points of matrix convex sets and their spanning properties
Eric Evert, Benjamin Passer, Tea Štrekelj
TL;DR
The paper surveys matrix convex sets as a noncommutative generalization of convex sets, focusing on diverse extreme points (matrix extreme, matrix exposed, free extreme) and their Krein–Milman-type results. It elucidates the Webster–Winkler theorem, density of matrix exposed points in matrix extremes, and spanning results for free spectrahedra, highlighting where finite-dimensional boundary representations suffice. It also contrasts real versus complex settings, the operator-system perspective, and the role of linear matrix inequalities in defining free spectrahedra and their semialgebraic geometry. The work clarifies when free extreme points provide minimal spanning sets and when they fail to exist, illustrating both the power and limitations of dimension-free convexity in operator algebra and NC polynomial contexts.
Abstract
This expository article gives a survey of matrix convex sets, a natural generalization of convex sets to the noncommutative (dimension-free) setting, with a focus on their extreme points. Mirroring the classical setting, extreme points play an important role in matrix convexity, and a natural question is, ``are matrix convex sets the (closed) matrix convex hull of their extreme points?" That is, does a Krein-Milman theorem hold in this setting? This question requires some care, as there are several notions of extreme points for matrix convex sets. Three of the most prevalent notions are matrix extreme points, matrix exposed points, and free extreme points. For each of these types of extreme points, we examine strengths and shortcomings in terms of a Krein-Milman theorem. Of particular note is the fact that these extreme points are all finite-dimensional in nature. As such, a large amount of our discussion is about free spectrahedra, which are matrix convex sets determined by a linear matrix inequality.