Uniqueness of completions and related topics |
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Authors: | Chan He Horst Martini Senlin Wu |
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Affiliation: | 1.Department of Mathematics,North University of China,Taiyuan,China;2.Faculty of Mathematics,Chemnitz University of Technology,Chemnitz,Germany |
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Abstract: | A bounded subset of a normed linear space is said to be (diametrically) complete if it cannot be enlarged without increasing the diameter. A complete super set of a bounded set K having the same diameter as K is called a completion of K. In general, a bounded set may have different completions. We study normed linear spaces having the property that there exists a nontrivial segment with a unique completion. It turns out that this property is strictly weaker than the property that each complete set is a ball, and it is strictly stronger than the property that each set of constant width is a ball. Extensions of this property are also discussed. |
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