On a Functional Operation Generating Convex Functions, Part 1: Duality

Abstract

The function $f \Delta g : (x, y) \mapsto g(y) f (x/g(y))$, $y \in$ dom g, is jointly convex provided f is convex and nonpositive at the origin and provided g is concave and nonnegative on its effective domain. Its convex conjugate combines the convex conjugates of f and −g by means of the same composition law. The effective domain of f Δg is then studied, which will prove to be useful in Part 2 of this paper (algebraic properties, Ref. 1).