Absolute extreme points of matrix convex sets: existence and spanning

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Stephens, Alexander
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The study of matrix convex sets is a relatively young endeavour, being first proposed in 1984 by Wittstock. Matrix convex sets can be found as the matrix state spaces of operator systems, and as the solution sets of linear matrix inequalities suitably modified to allow for matrix solutions. Some concepts from classical convexity can be translated to the matrix convexity setting in more than one way, such as the notion of an extreme point of a convex set. We will see two kinds of extreme point for matrix convex sets: one called a matrix extreme point, and the other called an absolute extreme point. Important results from classical convexity theory, such as the Krein-Milman theorem, have been proven to hold in the matrix convexity setting. One such Krein-Milman-type theorem was proven by Webster and Winkler in 1999, using the notion of a matrix extreme point. In this work, we present the proof of another Krein-Milman-type theorem, first proven by Evert and Helton in 2019, that holds for a special class of compact matrix convex sets, and that uses the notion of absolute extreme points. After this, we proceed to use absolute extreme points to propose a definition of strict matrix convexity, which as yet has no agreed upon definition.
matrix convexity, mathematics, extreme points, Krein-Milman theorem, strict convexity, free spectrahedra