Support functions and analytic inequalities

Abstract

The purpose of this paper is to provide an exposition of the theory of inequalities which connect convex functions by the method of support functions. The application of support functions is known [2] to be fruitful in the modern theory of extremal problems. At the same time there is no coherent and systematic application of support functions to such ancient and elementary extremal problems as inequalities. Two methodological purposes are pursued in this paper: On one hand, to illustrate the fertility of the method of separation of convex sets by the elementary material of inequalities; on the other hand to provide a systematic exposition of the theory of inequalities which connect convex functions by means of the apparatus of support functions. We mainly consider classical inequalities: the Minkowski and Holder inequalities, the arithmetic-geometric mean inequality, inequalities for elementary symmetric functions, determinantal inequalities and inequalities for indefinite forms. All these inequalities may be divided into two classes according to their geometric meaning. The inequalities of the first class mean geometrically that some cone is convex. The inequalities of the second class mean that the tangent hyperplane to the cone at some point is a supporting plane of the cone. The cone will always be an epigraph of a convex (or subgraph of a concave) positively homogeneous function f(x) and the corresponding inequalities will have the form