Kreĭn–Mil'man theorem for homogeneous polynomials

This note is devoted to the problem of recovering a convex set of homogeneous polynomials from the subset of its extreme points, i. e. to the justification of a polynomial version of the classical Kreĭn–Mil'man theorem. Not much was done in this direction. The existing papers are mostly devoted to 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 Kreĭn–Mil'man theorem does not work, since closed convex sets of operators turn out to be compact in some natural topology only in very 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 and an operator form of the Kreĭn–Mil'man theorem was obtained. Combining the mentioned approach with the homogeneous polynomials linearization, in this paper we obtain a version of the Kreĭn–Mil'man theorem for homogeneous polynomials. Namely, a weakly order bounded, operator convex and pointwise order closed set of homogeneous polynomials acting from an arbitrary vector space into Kantorovich space is the closure under pointwise order convergence of the operator convex hull of its extreme points. The Mil'man's inverse of the Kreĭn–Mil'man theorem for homogeneous polynomials is also established: The extreme points of the smallest operator convex pointwise order closed set containing a given set $A$ of homogeneous polynomials are pointwise uniform limits of appropriate mixings nets in $A$. The mixing of a family of polynomials with values in a Kantorovich space is understood as the (infinite) sum of these polynomials, multiplied by pairwise disjoint band projections with identity sum.