Generalized outer linearizations and extremal properties of rotational epi-symmetrizations

Abstract

We develop a functional extension of an extremal principle by Schneider (Monatsh. Math., 1967) by introducing generalized outer linearizations of convex functions. Given a coercive convex function on $\mathbb{R}^n$, a generalized outer linearization is defined as a convex minorant represented by a general but function-dependent set of slopes, thereby extending classical outer representations of convex bodies by supporting halfspaces. This representation converts geometric outer approximations by supporting halfspaces into functional approximations by supporting affine functions, and replaces outer normal data by a dual sampling problem in the domain of the Legendre--Fenchel transform. On a standard class of coercive convex functions, we derive a general extremal principle, showing that the rotational epi-symmetrization maximizes best approximations under outer linearizations of any monotone, concave functional that is upper semicontinuous with respect to epi-convergence. A central feature of the analysis is that it is carried out in the natural class of coercive, but not necessarily super-coercive, convex functions. Working in this setting introduces intricate topological and variational difficulties, which are addressed using refined duality and epi-convergence arguments. As an application of our main results, we derive a functional version of Urysohn's inequality, as well as an analytic extension of a classical covering result of Firey and Groemer (J. London Math. Soc., 1964). Finally, we prove an extremal inequality related to the piecewise affine approximation of convex functions.