Geometry of convex sets in spaces of homogeneous polynomials

The necessary and sufficient conditions under which the cone of positive homogeneous polynomials between vector lattices coincides with the closed convex hull of the set of sums of monomials in lattice homomorphisms are found. Incidentally the answer for the following question is established: when the cone of positive multilinear operators between vector lattices serves as a point-wise uniformly closed convex hull of the set of lattice multimorphisms.