Abstract. The aim of this paper is to make an attempt to justify the main results from Convex Analysis by one elegant tool, the conification method, which consists of three steps: conify, work with convex cones, deconify. It is based on the fact that the standard operations and facts (`the calculi') are much simpler for special convex sets, convex cones. By considering suitable classes of convex cones, we get the standard operations and facts for all situations in the complete generality that is required. The main advantages of this conification method are that the standard operations---linear image, inverse linear image, closure, the duality operator, the binary operations and the inf-operator---are defined on all objects of each class of convex objects---convex sets, convex functions, convex cones and sublinear functions---and that moreover the standard facts---such as the duality theorem---hold for all closed convex objects. This requires that the analysis is carried out in the context of convex objects over cosmic space, the space that is obtained from ordinary space by adding a horizon, representing the directions of ordinary space.