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Archiv der Mathematik

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

Uniqueness of completions and related topics

  • Chan He
  • Horst Martini
  • Senlin Wu
Article
  • 48 Downloads

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.

Keywords

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

Mathematics Subject Classification

46B20 52A10 

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References

  1. 1.
    J. Alonso, H. Martini, and S. Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math. 83 (2012), 153–189MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405–439MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Baronti and P.L. Papini, Diameters, centers and diametrically maximal sets, IVth Italian Conference on Integral Geometry, Geometric Probability Theory and Convex Bodies (Italian) (Bari, 1994), Rend. Circ. Mat. Palermo (2) Suppl. No. 38 (1995), 11–24.Google Scholar
  4. 4.
    L. Caspani and P.L. Papini, On constant width sets in Hilbert spaces and around, J. Convex Anal. 22 (2015), 889–900.MathSciNetzbMATHGoogle Scholar
  5. 5.
    G.D. Chakerian and H. Groemer, Convex bodies of constant width. In: Convexity and its Applications, 49–96, Birkhäuser, Basel, 1983.Google Scholar
  6. 6.
    W.J. Davis, A characterization of \(P_{1}\) spaces, J. Approximation Theory 21 (1977), 315–318.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    H.G. Eggleston, Sets of constant width in finite dimensional Banach spaces, Israel J. Math. 3 (1965), 163–172.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    E.L. Fuster, Some moduli and constants related to metric fixed point theory, In: Handbook of Metric Fixed Point Theory, 133–175, Kluwer Acad. Publ., Dordrecht, 2001.Google Scholar
  9. 9.
    E. Heil and H. Martini, Special convex bodies, In: Handbook of Convex Geometry, Vol. A, B, 347–385, North-Holland, Amsterdam, 1993.Google Scholar
  10. 10.
    H. Martini, C. Richter, and M. Spirova, Intersections of balls and sets of constant width in finite-dimensional normed spaces, Mathematika 59 (2013), 477–492.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    H. Martini and K.J. Swanepoel, The geometry of Minkowski spaces—a survey. II, Expo. Math. 22 (2004), 93–144.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    J.P. Moreno and R. Schneider, Diametrically complete sets in Minkowski spaces, Israel J. Math. 191 (2012), 701–720.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    L. Nachbin, A theorem of the Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950), 28–46.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    P.L. Papini, Complete sets and surroundings, In: Banach and Function Spaces IV (ISBFS 2012), 149–163, Yokohama Publ., Yokohama, 2014.Google Scholar
  15. 15.
    P.L. Papini, Completions and balls in Banach spaces, Ann. Funct. Anal. 6 (2015), 24–33.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    C.M. Petty, Equilateral sets in Minkowski spaces, Proc. Amer. Math. Soc. 29 (1971), 369–374.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    K.J. Swanepoel, Equilateral sets in finite-dimensional normed spaces, In: Seminar of Mathematical Analysis, Volume 71 of Colecc. Abierta, 195–237, Univ. Sevilla Secr. Publ., Seville, 2004.Google Scholar
  18. 18.
    D. Yost. Irreducible convex sets, Mathematika 38 (1991), 134–155.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© 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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