The Krein–Milman Theorem for Homogeneous Polynomials

Abstract

This note addresses the problem of recovering a convex set of homogeneous polynomials from the subset of its extreme points, i.e., the justification of a polynomial version of the classical Krein–Milman theorem. Not much was done in this direction. The existing papers deal mostly with the description of the extreme points of the unit ball in the space of homogeneous polynomials in various special cases. Even in the case of linear operators, the classical Krein–Milman theorem does not work, since closed convex sets of operators turn out to be compact in some natural topology only in rather special cases. In the 1980s, a new approach to the study of the extremal structure of convex sets of linear operators was proposed on the basis of the theory of Kantorovich spaces, which led to an operator form of the Krein–Milman theorem. Combining the approach with the linearization method for homogeneous polynomials, we obtain a version of the Krein–Milman theorem for homogeneous polynomials. Namely, a weakly order bounded, operator convex, and pointwise order closed set \( \Omega \) of homogeneous polynomials from an arbitrary vector space to a Kantorovich space is the pointwise order closure of the operator convex hull of the extreme points of \( \Omega \). We also establish Milman’s converse of the Krein–Milman theorem for homogeneous polynomials: The extreme points of the smallest operator convex pointwise order closed set including a given set \( \Omega \) of homogeneous polynomials are pointwise uniform limits of appropriate nets of mixings in \( \Omega \). A mixing of a family of polynomials with the values in a Kantorovich space is understood as the (infinite) sum of these polynomials multiplied by pairwise disjoint order projections with sum the identity operator in the Kantorovich space.