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

Extreme Points of Polar Convex Sets

Approximating Extreme Points of Infinite Dimensional Convex Sets

The property that an optimal solution to the problem of minimizing a continuous concave function over a compact convex set in IRn is attained at an extreme point is generalized by the Bauer Minimum Principle to the infinite dimensional context. The problem of approximating and characterizing infinite dimensional extreme points thus becomes an important problem. Consider now an infinite dimensio...

Extreme Points in Convex Sets of Symmetric Matrices

This paper deals with the following problem: What are the extreme points of a convex set K of n X n matrices, which is the intersection of the set S„ of symmetric matrices of nonnegative type, with another convex subset of symmetric matrices HI In the case where the facial structure of H is known, we expose a general method to determine the extreme points of K (Theorem 1). Then, we apply this m...

Matrix Convex Hulls of Free Semialgebraic Sets

This article resides in the realm of the noncommutative (free) analog of real algebraic geometry – the study of polynomial inequalities and equations over the real numbers – with a focus on matrix convex sets C and their projections Ĉ. A free semialgebraic set which is convex as well as bounded and open can be represented as the solution set of a Linear Matrix Inequality (LMI), a result which s...

POINTS FOR m-CONVEX SETS

Let S be closed, m-convex subset of R d S locally a full ddimensional, with Q the corresponding set of Inc points of S If q is an essential inc point of order k then for some neighborhood U of q Q u is expressible as a union of k or fewer (d2)-dimenslonal manifolds, each containing q For S compact, if to every q E Q there corresponds a k > 0 such that q is an essential inc point of order k then...