abstract convexity in measure theory and in convex analysis: Abstract convexity in measure theory and in convex analysis.
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| Title: | abstract convexity in measure theory and in convex analysis: Abstract convexity in measure theory and in convex analysis. |
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| Authors: | V. L. Levin |
| Source: | Journal of Mathematical Sciences. 116(4):3432-3467 |
| Publisher Information: | Springer US, New York, NY, 2003. |
| Publication Year: | 2003 |
| Subject Terms: | Programming in abstract spaces, Convex programming, abstract convexity, Functional analytic lifting theory, Lifting theory, Nonsmooth analysis, Variational problems in a geometric measure-theoretic setting, mass transportation, lifting |
| Description: | 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. |
| Document Type: | Article |
| File Description: | application/xml |
| ISSN: | 1072-3374 |
| DOI: | 10.1023/a:1024033523973 |
| Access URL: | https://zbmath.org/2094859 https://doi.org/10.1023/a:1024033523973 https://rd.springer.com/article/10.1023/a:1024033523973 http://mathecon.cemi.rssi.ru/vlevin/files/JMS2003.pdf https://link.springer.com/article/10.1023%2FA%3A1024033523973 |
| Accession Number: | edsair.dedup.wf.002..9d87ce636ad30f21e416cf7b3b1fc5c4 |
| Database: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | 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. |
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| ISSN: | 10723374 |
| DOI: | 10.1023/a:1024033523973 |