Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T15:00:50.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous Finite Apollonius Sets in Metric Spaces

Published online by Cambridge University Press:  20 November 2018

L. D. Loveland
L. D. Loveland*
Affiliation:
Utah State University, Logan, Utah
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The set of all points in the Euclidean plane E2, the ratio of whose distances from two fixed points is a constant ƛ, is known as the circle of Apollonius [7, p. 62]. This ‘'Apollonius” set is a circle except for the degenerate cases where ƛ = 1 or ƛ = 0. In more general metric spaces the same definition applies to select certain Apollonius sets (or “ ƛ-sets” in our terminology), but of course these sets are not always circles. For example, all ƛ-sets (ƛ > 0) relative to a circle in E2 are two-point sets, and all ƛ-sets relative to E1 are either singletons or two-point sets. This paper deals with the topological structure of a metric space when certain cardinality conditions have been imposed on its ƛ-sets.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 2 , 01 April 1976 , pp. 341 - 347
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Berard, Anthony D., Jr., Characterizations of metric spaces by use of their midsets: Intervals, Fund. Math. 73 (1971), 1–7.Google Scholar
2. Berard, Anthony D., Jr. and Nitka, W., A new definition of the circle by use of bisectors, Fund. Math. 84 (1974), 49–55.Google Scholar
3. Loveland, L. D., A metric characterization of a simple closed curve, to appear, General Topology and Appl.Google Scholar
4. Loveland, L. D. and Valentine, J. E., Convex metric spaces with C-dimensional midsets, Proc. Amer. Math. Soc. 37 (1973), 569–571.Google Scholar
5. Loveland, L. D. and Valentine, J. E., Characterizing a circle with the double midset property, Proc. Amer. Math. Soc. 53 (1975), 443–444.Google Scholar
6. Loveland, L. D. and Way ment, S. G., Characterizing a curve with the double midset property, Amer. Math. Monthly 51 (1974), 1003–1006.Google Scholar
7. Miller, Leslie H., College Geometry (Appleton-Century-Crofts, New York, 1957).Google Scholar
8. Moore, R. L., Foundations of point set theory (Amer. Math. Colloq. Publ. 13, 1962).Google Scholar
9. Willard, Stephen, General topology (Addison-Wesley, Reading, Mass., 1970).Google Scholar