Abstract

The classic notions of betweenness and equidistance in Euclidean geometry readily generalize to the context of metric spaces. We view these notions from an axiomatic perspective, then analyze the role various axioms play when interpreted in a metric space – especially one where the metric is induced by a vector space norm. The points lying between two given points constitute the betweenness interval bracketed by those points, and the points equidistant from two given points constitute the equiset with those points as cocenters. An equiset gives rise to a division of the underlying space into two comparative nearness regions; in the case of the Euclidean plane, each such region is a half-plane bounded by the line that is the equiset. Betweenness intervals naturally engender a notion of convexity, and one focus of this investigation is the issue of when equisets and their comparative nearness regions, as well as the betweenness intervals themselves, are convex. For normed vector spaces, betweenness intervals are always convex when the dimension is at most two, but this convexity property easily fails in higher dimensions. Equisets and comparative nearness regions in a normed vector space are convex precisely when the norm arises from an inner product. This is one of several characterizations we present of normed vector space properties purely in terms of abstract betweenness, equidistance and comparative nearness.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Betweenness
  • equidistance
  • comparative nearness
  • convexity
  • IR-axioms
  • metric spaces
  • normed vector spaces
  • taut arc-connectedness

Mathematics Subject Classification

51F99; 46B20, 52A01, 52A10, 52A21, 52A30, 54A05, 54E35

References

  1. D. Anderson, P. Bankston, A. McCluskey: Convexity in topological betweenness structures, Top. Appl. 304 (2021), art. 107783, 20 p.https://doi.org/10.1016/j.topol.2021.107783Google Scholar
  2. P. Bankston: Road systems and betweenness, Bull. Math. Sci. 3 (2013) 389–408.https://doi.org/10.1007/s13373-013-0040-4Google Scholar
  3. P. Bankston: The antisymmetry betweenness axiom and Hausdorff continua, Top. Proc. 45 (2015) 189–215.Google Scholar
  4. P. Bankston, A. McCluskey, R. J. Smith: Semicontinuity of betweenness functions, Topol. Appl. 246 (2018) 22–47.https://doi.org/10.1016/j.topol.2018.07.002Google Scholar
  5. A. R. Berard, Jr.: Characterizations of metric spaces by use of their midsets: Intervals, Fund. Math. 73 (1971) 1–7.https://doi.org/10.4064/fm-73-1-1-7Google Scholar
  6. N. Bourbaki: Topological Vector Spaces, Chapters 1–5, Springer, New York (2003).https://doi.org/10.1007/978-3-642-61715-7_1Google Scholar
  7. C. C. Chang, H. J. Keisler: Model Theory, 3rd ed., North Holland, Amsterdam (1990).Google Scholar
  8. M. M. Day: Some characterizations of inner-product spaces, Trans. Amer. Math. Soc. 62/2 (1947) 320–337.https://doi.org/10.1090/S0002-9947-1947-0022312-9Google Scholar
  9. T. Jahn, M. Spirova: On bisectors in normed planes, Contrib. Discrete Math. 10/2 (2015) 1–9.https://doi.org/10.55016/ojs/cdm.v10i2.62371Google Scholar
  10. K. Kuratowski: Topology, Vol. II, Academic Press, New York (1968).Google Scholar
  11. L. D. Loveland: When midsets are manifolds, Proc. Amer. Math. Soc. 61 (1976) 353–360.https://doi.org/10.1090/S0002-9939-1976-0438342-2Google Scholar
  12. L. D. Loveland: Metric spaces with connected midsets, Houston J. Math. 3/4 (1977) 495–501.Google Scholar
  13. H. Martini, K. J. Swanepoel: The geometry of Minkowski spaces – a survey. Part II, Expo. Math. 22 (2004) 93–144.https://doi.org/10.1016/S0723-0869(04)80009-4Google Scholar
  14. A. McCluskey, B. McMaster: Undergraduate Topology (A Working Textbook), Oxford University Press, Oxford (2014).Google Scholar
  15. K. Menger: Untersuchungen über allgemeine Metrik, Math. Ann. 100 (1928) 75–163.https://doi.org/10.1007/BF01448840Google Scholar
  16. S. B. Nadler, Jr.: An embedding theorem for certain spaces with an equidistant property, Proc. Amer. Math. Soc. 59 (1976) 179–183.https://doi.org/10.1090/S0002-9939-1976-0410686-XGoogle Scholar
  17. S. B. Nadler, Jr.: Continuum Theory, Marcel Dekker, New York (1992).Google Scholar
  18. B. B. Panda, O. P. Kapoor: On equidistant sets in normed linear spaces, Bull. Austral. Math. Soc. 11 (1974) 443–454.https://doi.org/10.1017/S0004972700044075Google Scholar

Suggested citation

P. Bankston, A. McCluskey. “On Betweenness and Equidistance in Metric Spaces.” Journal of Convex Analysis 30 (2023), No. 1, 371–400. https://doi.org/10.68381/jca30021

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.