2024/12/17 by Igor Klep, Scott McCullough, Klep, Igor +3 · 1 citation
Computer Science · Mathematics · #Advanced Banach Space Theory #Combinatorics #Convexity #Duality (order theory) #Economics #Extreme point #Financial economics #Geology #Hull #Mathematical Inequalities and Applications #Mathematical economics #Mathematics #Noncommutative geometry #Optimization and Variational Analysis #Pure mathematics #math.FA #math.OA
paper · pdf · doi:10.48550/arxiv.2412.13267
openalex publication_date 2024/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article studies generalizations of (matrix) convexity, including partial convexity and biconvexity, under the umbrella of Γ-convexity. Here Γ is a tuple of free symmetric polynomials determining the geometry of a Γ-convex set. The paper introduces the notions of Γ-operator systems and Γ-ucp maps and establishes a Webster-Winkler type categorical duality between Γ-operator systems and Γ-convex sets. Next, a notion of an extreme point for Γ-convex sets is defined, paralleling the concept of a free extreme point for a matrix convex set. To ensure the existence of such points, the matricial sets considered are extended to include an operator level. It is shown that the Γ-extreme points of an operator Γ-convex set K are in correspondence with the free extreme points of the operator convex hull of Γ(K). From this result, a Krein-Milman theorem for Γ-convex sets follows. Finally, relying on the results of Helton and the first two authors, a construction of an approximation scheme for the Γ-convex hull of the matricial positivity domain (also known as a free semialgebraic set) Dp of a free symmetric polynomial p is given. The approximation consists of a decreasing family of Γ-analogs of free spectrahedra, whose projections, under mild assumptions, in the limit yield the Γ-convex hull of Dp.