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

Our main intention 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 in these classes. One main example which we discovered is a new duality transform for convex non-negative functions on ℝⁿ attaining the value 0 at the origin (which we call "geometric convex functions").¹ 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.