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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Senlin Wu is supported by the National Natural Science Foundation of China (Grant Nos. 11371114 and 11571085), and is the corresponding author.
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He, C., Martini, H. & Wu, S. Uniqueness of completions and related topics. Arch. Math. 111, 157–165 (2018). https://doi.org/10.1007/s00013-018-1177-x
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DOI: https://doi.org/10.1007/s00013-018-1177-x