A simple view on convex analysis and its applications

Our aim is to give a simple view on the basics and applications of convex analysis. The essential feature of this account is the systematic use of the possibility to associate to each convex object---such as a convex set, a convex function or a convex extremal problem--- a cone, without loss of information. The core of convex analysis is the possibility of the dual description of convex objects, geometrical and algebraical, based on the duality of vectorspaces; for each type of convex objects, this property is encoded in an operator of duality, and the name of the game is how to calculate these operators. The core of this paper is a unified presentation, for each type of convex objects, of the duality theorem and the complete list of calculus rules.\nNow we enumerate the advantages of the `cone'-approach. It gives a unified and transparent view on the subject. The intricate rules of the convex calculus all flow naturally from one common source. We have included for each rule a precise description of the weakest convenient assumption under which it is valid. This appears to be useful for applications; however, these assumptioons are usually not given. We explain why certain convex objects have to be excluded in the definition of the operators of duality: the collections of associated cones of the target of an operator of duality need not be closed (here `closed' is meant in an algebraic sense). This makes clear that the remedy is to take the closure of the target. As a byproduct of the cone approach, we have found the solution of the open problem of how to use the polar operation to give a dual description of arbitrary convex sets. \nThe approach given can be extended to the infinite-dimensional case.