On Polar Cones and Differentiability in Reflexive Banach Spaces
Let $X$ be a Banach space, $Csubset X$ be a closed convex set included in a well-based cone $K$, and also let $sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a bounded base for the cone $K$, then using the obtained results, we find...
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| Format: | Article |
| Language: | English |
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University of Maragheh
2018-08-01
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| Series: | Sahand Communications in Mathematical Analysis |
| Subjects: | |
| Online Access: | http://scma.maragheh.ac.ir/article_32215_2e744dde303f4e6c175af724da107e48.pdf |
| _version_ | 1829531078163955712 |
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| author | Ildar Sadeqi Sima Hassankhali |
| author_facet | Ildar Sadeqi Sima Hassankhali |
| author_sort | Ildar Sadeqi |
| collection | DOAJ |
| description | Let $X$ be a Banach space, $Csubset X$ be a closed convex set included in a well-based cone $K$, and also let $sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a bounded base for the cone $K$, then using the obtained results, we find some geometric conditions for the set $C$, so that ${mathop{rm int}}(mathrm{dom} sigma_C) neqemptyset$. The latter is a primary condition for subdifferentiability of the support function $sigma_C$. Eventually, we study Gateaux differentiability of support function $sigma_C$ on two sets, the polar cone of $K$ and ${mathop{rm int}}(mathrm{dom} sigma_C)$. |
| first_indexed | 2024-12-16T18:14:01Z |
| format | Article |
| id | doaj.art-af7ff0bad3a64cf3a18cff542c4ff282 |
| institution | Directory Open Access Journal |
| issn | 2322-5807 2423-3900 |
| language | English |
| last_indexed | 2024-12-16T18:14:01Z |
| publishDate | 2018-08-01 |
| publisher | University of Maragheh |
| record_format | Article |
| series | Sahand Communications in Mathematical Analysis |
| spelling | doaj.art-af7ff0bad3a64cf3a18cff542c4ff2822022-12-21T22:21:42ZengUniversity of MaraghehSahand Communications in Mathematical Analysis2322-58072423-39002018-08-01111132310.22130/scma.2018.72221.28432215On Polar Cones and Differentiability in Reflexive Banach SpacesIldar Sadeqi0Sima Hassankhali1Department of Mathematics, Faculty of Science, Sahand University of Technology, Tabriz, Iran.Department of Mathematics, Faculty of Science, Sahand University of Technology, Tabriz, Iran.Let $X$ be a Banach space, $Csubset X$ be a closed convex set included in a well-based cone $K$, and also let $sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a bounded base for the cone $K$, then using the obtained results, we find some geometric conditions for the set $C$, so that ${mathop{rm int}}(mathrm{dom} sigma_C) neqemptyset$. The latter is a primary condition for subdifferentiability of the support function $sigma_C$. Eventually, we study Gateaux differentiability of support function $sigma_C$ on two sets, the polar cone of $K$ and ${mathop{rm int}}(mathrm{dom} sigma_C)$.http://scma.maragheh.ac.ir/article_32215_2e744dde303f4e6c175af724da107e48.pdfRecession conePolar coneBounded baseSupport functionGateaux differentiability |
| spellingShingle | Ildar Sadeqi Sima Hassankhali On Polar Cones and Differentiability in Reflexive Banach Spaces Sahand Communications in Mathematical Analysis Recession cone Polar cone Bounded base Support function Gateaux differentiability |
| title | On Polar Cones and Differentiability in Reflexive Banach Spaces |
| title_full | On Polar Cones and Differentiability in Reflexive Banach Spaces |
| title_fullStr | On Polar Cones and Differentiability in Reflexive Banach Spaces |
| title_full_unstemmed | On Polar Cones and Differentiability in Reflexive Banach Spaces |
| title_short | On Polar Cones and Differentiability in Reflexive Banach Spaces |
| title_sort | on polar cones and differentiability in reflexive banach spaces |
| topic | Recession cone Polar cone Bounded base Support function Gateaux differentiability |
| url | http://scma.maragheh.ac.ir/article_32215_2e744dde303f4e6c175af724da107e48.pdf |
| work_keys_str_mv | AT ildarsadeqi onpolarconesanddifferentiabilityinreflexivebanachspaces AT simahassankhali onpolarconesanddifferentiabilityinreflexivebanachspaces |