The Krein–Milman Theorem for Homogeneous Polynomials

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 ofthe 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 compactin some natural topology only in rather special cases. In the 1980s, a new approach to the study of theextremal structure of convex sets of linear operators was proposed on the basis of thetheory 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 theKrein–Milman theorem for homogeneous polynomials.Namely, a weakly order bounded, operator convex, and pointwise order closed set  $ \Omega $ ofhomogeneous polynomials from an arbitrary vector space to a Kantorovich space isthe pointwise order closure of the operator convex hull of the extremepoints 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 setincluding a given set  $ \Omega $ of homogeneous polynomials are pointwise uniformlimits of appropriate nets of mixings in  $ \Omega $ .A mixing of a family of polynomials with thevalues in a Kantorovich space is understood as the (infinite) sum of these polynomialsmultiplied by pairwise disjoint order projections with sum the identity operatorin the Kantorovich space.