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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
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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

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