Abstract convexity in measure theory and in convex analysis.
The paper under report is a survey on the so-called ``abstract convex analysis'' and on some applications to optimization problems. If \(\Omega\) is a set and \(H\) is a class of functions from \(\Omega\) into \(\mathbb{R}\), a function \(f: \Omega\to\mathbb{R}\cup \{+\infty\}\) is called \(H\)-convex if it is the supremum of a family of functions belonging to \(H\). Similarly, a set \(M\subset\Omega\) is called \(H\)-convex if it is the intersection of a family of sublevels of functions belonging to \(H\). The properties of these abstract notions are investigated in detail and some applications are presented: the theory of liftings on subspaces of \({\mathcal L}^\infty\), the subdifferential of integral functionals, and the Monge-Kantorovich problem of optimal mass transportation. In particular, for the last problem, the existence and uniqueness of an optimal transport map is shown under slightly more general assumptions with respect to what was previously shown in the literature.
scientific article; zbMATH DE number 2094859