Extreme Points of Matrix Convex Sets, Free Spectrahedra, and Dilation Theory

Abstract

For matrix convex sets a unified geometric interpretation of notions of extreme points and of Arveson boundary points is given. These notions include, in increasing order of strength, the core notions of "Euclidean" extreme points, "matrix" extreme points, and "absolute" extreme points. A seemingly different notion, the "Arveson boundary", has by contrast a dilation theoretic flavor. An Arveson boundary point is an analog of a (not necessarily irreducible) boundary representation for an operator system. This article provides and explores dilation theoretic formulations for the above notions of extreme points. The scalar solution set of a linear matrix inequality (LMI) is known as a spectrahedron. The matricial solution set of an LMI is a free spectrahedron. Spectrahedra (resp. free spectrahedra) lie between general convex sets (resp. matrix convex sets) and convex polyhedra (resp. free polyhedra). As applications of our theorems on extreme points, it is shown the polar dual of a matrix convex set K is generated, as a matrix convex set, by finitely many Arveson boundary points if and only if K is a free spectrahedron; and if the polar dual of a free spectrahedron K is again a free spectrahedron, then at the scalar level K is a polyhedron.