Functional calculus and Minkowski duality on vector lattices

The paper extends homogeneous functional calculus on vector lattices. It is shown that the function of elements of a relatively uniformly complete vector lattice can naturally be defined if the positively homogeneous function is defined on some conic set and is continuous on some closed convex subcone. An interplay between Minkowski duality and homogeneous functional calculus leads to the envelope representation of abstract convex elements generated by the linear hull of a finite collection in a uniformly complete vector lattice.