Convexity in matrix spaces, extreme points and facesŠtrekelj, Tea (Avtor) Klep, Igor (Mentor) matrix convex setmatrix extreme pointmatrix exposed pointKrein-Milman theoremStraszewicz-Klee theoremmatrix facematrix exposed facefree spectrahedron$\Gamma$-convex set$\Gamma$-operator system$\Gamma$-ucp mapfree extreme point$\Gamma$-extreme pointThis thesis investigates the notions of exposed points and (exposed) faces in the matrix convex setting. Matrix exposed points in finite dimensions were first defined by Kriel in 2019. Here this notion is extended to matrix convex sets in infinite-dimensional vector spaces. Then a connection between matrix exposed points and matrix extreme points is established: a matrix extreme point is ordinary exposed if and only if it is matrix exposed. This leads to a Krein-Milman type result for matrix exposed points that is due to Straszewicz-Klee in classical convexity: a compact matrix convex set is the closed matrix convex hull of its matrix exposed points. Moreover, with similar techniques, an even stronger result is obtained, namely that the matrix exposed points are dense in the matrix extreme points. Several notions of a fixed-level as well as a multilevel matrix face and matrix exposed face are introduced to extend the concepts of a matrix extreme point and a matrix exposed point, respectively. Their properties resemble those of (exposed) faces in the classical sense, e.g., it is shown that the $C^\ast$-extreme (matrix extreme) points of a matrix face (matrix multiface) of a matrix convex set $K$ are matrix extreme in $K$. As in the case of extreme points, any fixed-level matrix face is ordinary exposed if and only if it is a matrix exposed face. From this it follows that every fixed-level matrix face of a free spectrahedron is matrix exposed. On the other hand, matrix multifaces give rise to the noncommutative counterpart of the classical theory connecting (archimedean) faces of compact convex sets and (archimedean) order ideals of the corresponding function systems. The final part of this thesis studies several generalizations of (matrix) convexity, e.g., partial convexity or biconvexity, which are summed up in the term $\Gamma$-convexity. Here $\Gamma$ is a tuple of symmetric free polynomials determining the geometry of a $\Gamma$-convex set. The notions of $\Gamma$-operator systems and $\Gamma$-ucp maps are introduced and a Webster-Winkler type categorical duality between $\Gamma$-operator systems and $\Gamma$-convex sets is established. Next, a notion of extreme points of $\Gamma$-convex sets is introduced so that it extends the concept of a free extreme point. To ensure that such points exist, matrix (but also $\Gamma$-) convex sets are extended to include an operator level. The existence of free extreme points of the operator convex hull of $\Gamma(K)$ then guarantees existence of the so called $\Gamma$-extreme points of an operator $\Gamma$-convex set $K$. This result is key to establish a Krein-Milman theorem for $\Gamma$-convex sets. Finally, relying on the results of Helton, Klep and McCullough, a construction of an approximation scheme for the $\Gamma$-convex hull of the matricial positivity domain of a symmetric free polynomial $p$ is given. The approximation consists of a decreasing family of $\Gamma$-analogs of free spectrahedra, which under mild assumptions captures the $\Gamma$-convex hull of the matricial positivity domain of $p$.20242024-10-10 11:45:49Doktorsko delo/naloga163763sl