Abstract convexity in measure theory and in convex analysis.
type: Item
instance of: scholarly article
title: Abstract convexity in measure theory and in convex analysis.
zbMATH Open document ID: 1074.28003
DOI: 10.1023/A:1024033523973
published in: Journal of Mathematical Sciences (New York)
publication date: 2004-08-24T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 28A51
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 46G15
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 2094859
zbMATH Keywords: abstract convexity
zbMATH Keywords: lifting
zbMATH Keywords: mass transportation
author: Vladimir L. Levin
reviewed by: Giuseppe Buttazzo
MaRDI profile type: Publication
full work available at URL: a:1024033523973
OpenAlex ID: W203736716
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description: scientific article; zbMATH DE number 2094859