Minkowski Duality and Its Applications

This article is an account of problems grouped around the concept of Minkowski duality - one of the central constructions in convex analysis. The article consists of an introduction, four sections, and a commentary.In § 1 we set out the main facts about H-convex elements and introduce the Minkowski-Fenchel and the Minkowski-Moreau schemes; we consider the space of H-convex sets. Here we collect together the main examples, namely the convex and sublinear functions, and the stable, normal, and convex sets in the sense of Fan, amongst others. § 2 is concerned mainly with representations of positive functionals over continuous H-convex functions and sets. Here we also establish the links between such constructions and the Choquet theory.In § 3 we introduce various characterizations of H-convexity in the form of theorems on supremal generators. In particular, we consider in detail theorems on the definability of convergence of sequences of operators in terms of their convergence on a cone. Other applications of supremal generators are also given.In § 4 problems of isoperimetric type (with an arbitrary number of constraints) in the geometry of convex surfaces are analyzed as problems of programming in a space of convex sets. We examine as particular examples exterior and interior isoperimetric problems, the Uryson problem, and others.