Hidden structures in the class of convex functions and a new duality transform

Abstract

Our main intension in this paper is to demonstrate how some seemingly purely geometric notions can be presented and understood in an analytic language of inequalities and then, with this understanding, can be defined for classes of functions and reveal new and hidden structures within these classes. One main example which we discovered is a new duality transform for convex non-negative functions on $R^n$ attaining thevalue $0$ at the origin (which we called “geometric convex functions”). This transform, together with the classical Legendre transform, are essentially the only existing duality relations on this class of functions. Using these dualities we show that the geometric constructions of Support and Minkowski–functional may be extended, in a unique way, to the class of geometric log-concave functions, revealing hidden geometric structures on this class of functions.