Envelopes and clutters

In this paper, a set function ϕ defined on a finite set Ω is said to be an upper envelope if there exists a set {pi} of nonnegative vectors on Ω such that ϕ(G) = max {p1(G),…,pn(G)} for all G ⊂ Ω. All upper envelopes form a convex cone. We give a necessary and sufficient condition for an upper envelope to be extremal in the cone of all upper envelopes in terms of its representation. Furthermore we study the upper envelopes represented by clutters. We show that a clutter is extremal in the cone of the upper envelopes if and only if it satisfies some kind of connectivity.