Best approximation in ℝ-trees
An ℝ-tree is a geodesic space for which there is a unique arc joining any two of its points, and this arc is a metric segment. We give a constructive proof of the following "best approximation" theorem in such spaces. Suppose X is a closed convex and geodesically bounded subset of an ℝ-tre...
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th-cmuir.6653943832-52652014-08-30T02:56:20Z Best approximation in ℝ-trees Kirk W.A. Panyanak B. An ℝ-tree is a geodesic space for which there is a unique arc joining any two of its points, and this arc is a metric segment. We give a constructive proof of the following "best approximation" theorem in such spaces. Suppose X is a closed convex and geodesically bounded subset of an ℝ-tree H, and suppose T:X2H is a multivalued upper semicontinuous mapping whose values are nonempty closed convex subsets of X. Then there exists a point x0X such that [image omitted] We also give a topological version of the above theorem in a more abstract setting, and we prove a KKM theorem for geodesically bounded ℝ-trees. 2014-08-30T02:56:20Z 2014-08-30T02:56:20Z 2007 Article 01630563 10.1080/01630560701348517 NFAOD http://www.scopus.com/inward/record.url?eid=2-s2.0-34249098113&partnerID=40&md5=fa3d6f01b2140fd5433b5e8adf14463d http://cmuir.cmu.ac.th/handle/6653943832/5265 English |
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An ℝ-tree is a geodesic space for which there is a unique arc joining any two of its points, and this arc is a metric segment. We give a constructive proof of the following "best approximation" theorem in such spaces. Suppose X is a closed convex and geodesically bounded subset of an ℝ-tree H, and suppose T:X2H is a multivalued upper semicontinuous mapping whose values are nonempty closed convex subsets of X. Then there exists a point x0X such that [image omitted] We also give a topological version of the above theorem in a more abstract setting, and we prove a KKM theorem for geodesically bounded ℝ-trees. |
| format |
Article |
| author |
Kirk W.A. Panyanak B. |
| spellingShingle |
Kirk W.A. Panyanak B. Best approximation in ℝ-trees |
| author_facet |
Kirk W.A. Panyanak B. |
| author_sort |
Kirk W.A. |
| title |
Best approximation in ℝ-trees |
| title_short |
Best approximation in ℝ-trees |
| title_full |
Best approximation in ℝ-trees |
| title_fullStr |
Best approximation in ℝ-trees |
| title_full_unstemmed |
Best approximation in ℝ-trees |
| title_sort |
best approximation in ℝ-trees |
| publishDate |
2014 |
| url |
http://www.scopus.com/inward/record.url?eid=2-s2.0-34249098113&partnerID=40&md5=fa3d6f01b2140fd5433b5e8adf14463d http://cmuir.cmu.ac.th/handle/6653943832/5265 |
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