a. If all δ-cuts of ψ (·) are finite unions of compact intervals then the convex hull of ψ (·) is ψ (·) itself. If some δ-cuts of ψ (·) are not compact or not finite unions of compact intervals then the function ξ(·) is defined by
A concrete example is the following:
The -cut of ψ (·) is (a; b] which is not closed. Therefore the convex hull ξ(·) is the following
b. From β < δ it follows Cβ(x*) ⊇ Cδ(x*) by definition of δ-cuts. Therefore we obtain
In order to prove the equality of both sets assume x ∈ Cβ(x*) for all β > δ. Then ξx*(x) ≥ β for all β < δ. Assuming ξx*(x) < δ yields a contradiction. Therefore and the equality follows.