Uniqueness of completions and related topics
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.
KeywordsBanach spaces Completeness Constant width set Normed linear space Uniqueness of completion
Mathematics Subject Classification46B20 52A10
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