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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
2014
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| Online Access: | 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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| Institution: | Chiang Mai University |
| Language: | English |
| Summary: | 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. |
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