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Metric Spaces in Which All Triangles Are Degenerate

Bettina Richmond, Thomas Richmond
1997 The American mathematical monthly  
In any subspace of the real line R with the usual Euclidean metric d(x, y) = |x − y|, every triangle is degenerate. In R 2 or R 3 with the usual Euclidean metrics, a triangle is degenerate if and only if its vertices are collinear. With our intuition of a degenerate triangle having "collinear vertices" extended to arbitrary metric spaces, we might expect that a metric space in which every triangle is degenerate must be "linear". It might be reasonable to expect that any "linear" metric space is
more » ... ar" metric space is isometric to a subset of R with the usual metric. When classifying all metric spaces that have only degenerate triangles, we find that there are such metric spaces other than (isometric images of) subspaces of R. These other spaces, however, all have precisely four points and all are of the same form. In the final section, we illustrate that the usual topology on R 2 can not be generated by any metric in which all triangles are degenerate. A metric space (M, ρ) is a set of points M with a metric, or distance function, ρ : M × M −→ [0, ∞) that satisfies some natural properties we expect of distances: ρ(x, y) = ρ(y, x) for any x, y ∈ M , ρ(x, y) = 0 if and only if x = y, and ρ(x, y) + ρ(y, z) ≥ ρ(x, z) for any x, y, z ∈ M (triangle inequality).
doi:10.2307/2975234 fatcat:zzij3gzcnncv3phchexpyeqzvu