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arXiv · 1802.09133

Uniqueness of completions and related topics

Chan He·Horst Martini·Senlin Wu
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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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Chan He, Horst Martini, Senlin Wu. 2018-02-26. Uniqueness of completions and related topics. https://arxiv.org/abs/1802.09133

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