-convex Riemannian submanifolds.
We give a characterization of convex hypersurfaces with an equichordal point in terms of hedgehogs of constant width.
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 attaining the value 0 at the origin (which we call “geometric convex functions”). This transform, together...