Geodesic
Continuous Finite Apollonius Sets in Metric Spaces
Canadian journal of mathematics, Tome 28 (1976) no. 2, pp. 341-347

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Loveland, L. D. Continuous Finite Apollonius Sets in Metric Spaces. Canadian journal of mathematics, Tome 28 (1976) no. 2, pp. 341-347. doi: 10.4153/CJM-1976-036-5
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