Real- and Complex-valued Functions

Abstract

A map ϕ: of a convex open subset K of a vector space V is convex iff for all t in [0,1] and any x and y in K

$$ \varphi (t{\text{x + (1 - }}t{\text{)y}}) \leqslant t\varphi ({\text{x}}) + (1 - t)\varphi ({\text{y}}) $$

. When K is an open interval in ℝ and a ∈ K, a line l through (a, φ(a)) is a supporting line iff l lies below the graph of φ: {(p, q) ∈ l} ⇒ {q ≤ φ(p)}.