Convexities and Existence of the Farthest Point
Five counterexamples are given, which show relations among the new convexities and some important convexities in Banach space. Under the assumption that Banach space 𝑋 is nearly very convex, we give a sufficient condition that bounded, weakly closed subset of 𝑋 has the farthest points. We also give...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/139597 |
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| _version_ | 1832550055747256320 |
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| author | Z. H. Zhang C. Y. Liu |
| author_facet | Z. H. Zhang C. Y. Liu |
| author_sort | Z. H. Zhang |
| collection | DOAJ |
| description | Five counterexamples are given, which show relations among the
new convexities and some important convexities in Banach space. Under the assumption that Banach space 𝑋 is nearly very convex, we give a sufficient
condition that bounded, weakly closed subset of 𝑋 has the farthest points. We also give a sufficient condition that the farthest point map is single valued in a
residual subset of 𝑋 when 𝑋 is very convex. |
| format | Article |
| id | doaj-art-364e842bf26449baba2a1422d04d4480 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-364e842bf26449baba2a1422d04d44802025-02-03T06:07:52ZengWileyAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/139597139597Convexities and Existence of the Farthest PointZ. H. Zhang0C. Y. Liu1College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201600, ChinaCollege of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201600, ChinaFive counterexamples are given, which show relations among the new convexities and some important convexities in Banach space. Under the assumption that Banach space 𝑋 is nearly very convex, we give a sufficient condition that bounded, weakly closed subset of 𝑋 has the farthest points. We also give a sufficient condition that the farthest point map is single valued in a residual subset of 𝑋 when 𝑋 is very convex.http://dx.doi.org/10.1155/2011/139597 |
| spellingShingle | Z. H. Zhang C. Y. Liu Convexities and Existence of the Farthest Point Abstract and Applied Analysis |
| title | Convexities and Existence of the Farthest Point |
| title_full | Convexities and Existence of the Farthest Point |
| title_fullStr | Convexities and Existence of the Farthest Point |
| title_full_unstemmed | Convexities and Existence of the Farthest Point |
| title_short | Convexities and Existence of the Farthest Point |
| title_sort | convexities and existence of the farthest point |
| url | http://dx.doi.org/10.1155/2011/139597 |
| work_keys_str_mv | AT zhzhang convexitiesandexistenceofthefarthestpoint AT cyliu convexitiesandexistenceofthefarthestpoint |