More on Functions

Let Ω ⊆ RN be a domain and let F be a family of real-valued functions on Ω (we do not assume in advance that F is closed under any algebraic operations, although often in practice it will be). Let K be a compact subset of Ω. Then the convex hull of K in Ω with respect to F is defined to be K̂F ≡ { x ∈ Ω : f(x) ≤ sup t∈K f(t) for all f ∈ F } .