Archiv der Mathematik

, Volume 111, Issue 2, pp 157–165

# Uniqueness of completions and related topics

• Chan He
• Horst Martini
• Senlin Wu
Article

## 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

46B20 52A10

## References

1. 1.
J. Alonso, H. Martini, and S. Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math. 83 (2012), 153–189
2. 2.
N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405–439
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.
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.
7. 7.
H.G. Eggleston, Sets of constant width in finite dimensional Banach spaces, Israel J. Math. 3 (1965), 163–172.
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.
11. 11.
H. Martini and K.J. Swanepoel, The geometry of Minkowski spaces—a survey. II, Expo. Math. 22 (2004), 93–144.
12. 12.
J.P. Moreno and R. Schneider, Diametrically complete sets in Minkowski spaces, Israel J. Math. 191 (2012), 701–720.
13. 13.
L. Nachbin, A theorem of the Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950), 28–46.
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.
16. 16.
C.M. Petty, Equilateral sets in Minkowski spaces, Proc. Amer. Math. Soc. 29 (1971), 369–374.
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.

© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Chan He
• 1
• Horst Martini
• 2
• Senlin Wu
• 1
1. 1.Department of MathematicsNorth University of ChinaTaiyuanChina
2. 2.Faculty of MathematicsChemnitz University of TechnologyChemnitzGermany