Skip to main content
SpringerLink
Log in
Book cover

General Inequalities 3 pp 307–320Cite as

Birkhäuser

Menger-Betweenness in α-Simple Spaces

  • Chapter

  • 298 Accesses

Abstract

Let (S,d,G;α) be an α-simpie space with α > 1, and let B(p,r) be the set of all points q in S which are Menger-between p and r, together with p and r. In this paper, we obtain best possible upper and lower bounds for B(p,r). Furthermore, we show that if (S, ∥ · ∥) is a normed linear space and d(p,q) = ∥p − q∥, then B(p,r) is convex and p, r are on the boundary of B(p,r), but that this need not be the case when the metric d is not derived from a norm.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Blanc, Les espaces métriques quasi convexes. Annales scientifiques de l’École Normale Supérieure, Troisième Série 55 (1938).

    Google Scholar 

  2. L. M. Blumenthal, Theory and Applications of Distance Geometry. Oxford University Press, 1953.

    Google Scholar 

  3. M. Kuczma, Functional Equations in a Single Variable. Monografie Matematyczne, V. 46, Warsaw, 1960.

    Google Scholar 

  4. K. Menger, Statistical metrics. Proc. Nat. Acad. Sci. U. S. A. 28 (1942), 535–537.

    Article  Google Scholar 

  5. R. Moynihan and B. Schweizer, Betweenness relations in probabilistic metric spaces. Pacific J. Math. 81 (1979), 175–196.

    Article  Google Scholar 

  6. B. Schweizer and A. Sklar, Triangle inequalities in a class of statistical metric spaces. J. London Math. Soc. 38 (1963), k01–k06.

    Google Scholar 

  7. B. Schweizer and A. Sklar, Probabilistic Metric Spaces. Elsevier North Holland, New York, 1982.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département Mathemàtiques E. A. V., Universitat Politècnica de Barcelona, Barcelona 28, Spain

    C. Alsina

  2. Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts, 01003, USA

    B. Schweizer

Authors
  1. C. Alsina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Schweizer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of California, Los Angeles, California, USA

    E. F. Beckenbach

  2. Mathematisches Institut I, Universität Karlsruhe, Kaiserstraße 12, D-7500, Karlsruhe 1, Germany

    Wolfgang Walter

Rights and permissions

Reprints and permissions

Copyright information

© 1983 Springer Basel AG

About this chapter

Cite this chapter

Alsina, C., Schweizer, B. (1983). Menger-Betweenness in α-Simple Spaces. In: Beckenbach, E.F., Walter, W. (eds) General Inequalities 3. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 64. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6290-5_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-6290-5_24

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-6292-9

  • Online ISBN: 978-3-0348-6290-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation