2024/05/13 by Eric Evert, Evert, Eric, Benjamin Passer +3 · 2 citations
Computer Science · Mathematics · #13J30 #46N10 #47A20 #47L07 #47L25 #90C22 #Advanced Optimization Algorithms Research #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2405.07924
openalex publication_date 2024/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.