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Matrix convexity generalizes convexity to the dimension free setting and hasconnections to many mathematical and applied pursuits including operatortheory, quantum information, noncommutative optimization, and linear controlsystems. In the setting of classical convex sets, extreme points are centralobjects which exhibit many important properties. For example, the Minkowskitheorem shows that any element of a closed bounded convex set can be expressedas a convex combination of extreme points. Extreme points are also of greatinterest in the dimension free setting of matrix convex sets; however, here thesituation requires more nuance.  In the dimension free setting, there are many different types of extremepoints. Of particular importance are free extreme points, a highly restrictedtype of extreme point that is closely connected to the dilation theoreticArveson boundary. If free extreme points span a matrix convex set throughmatrix convex combinations, then they satisfy a strong notion of minimality indoing so. However, not all closed bounded matrix convex sets even have freeextreme points. Thus, a major goal is to determine which matrix convex sets arespanned by their free extreme points.  Building on a recent work of J. W. Helton and the author which shows thatfree spectrahedra, i.e., dimension free solution sets to linear matrixinequalities, are spanned by their free extreme points, we establish twoadditional classes of matrix convex sets which are the matrix convex hull oftheir free extreme points. Namely, we show that closed bounded freespectrahedrops, i.e, closed bounded projections of free spectrahedra, are thespan of their free extreme points. Furthermore, we show that if one considerslinear operator inequalities that have compact operator defining tuples, thenthe resulting ``generalized" free spectrahedra are spanned by their freeextreme points.

Free extreme points span generalized free spectrahedra given by compact coefficients