Archiv der Mathematik

, Volume 111, Issue 2, pp 157–165 | Cite as

Uniqueness of completions and related topics

  • Chan He
  • Horst Martini
  • Senlin Wu


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.


Banach spaces Completeness Constant width set Normed linear space Uniqueness of completion 

Mathematics Subject Classification

46B20 52A10 


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNorth University of ChinaTaiyuanChina
  2. 2.Faculty of MathematicsChemnitz University of TechnologyChemnitzGermany

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