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(m,n)-Equidistant Sets in \(\mathbb{R}^{k},\mathbb{S}^{k}\) , and ℙk

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Abstract

We call a metric space X (m,n)-equidistant if, when AX has exactly m points, there are exactly n points in X each of which is equidistant from (the points of) A. We prove that, for k≥2, the Euclidean space ℝk contains an (m,1)-equidistant set if and only if km. Although the sphere \(\mathbb{S}^{2}\) is (3,2)-equidistant, \(\mathbb{S}^{3}\) and ℝ4 contain no (4,2)-equidistant sets. We discuss related results about projective spaces, and state a conjecture about \(\mathbb{S}^{2}\) analogous to the Double Midset Conjecture.

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Correspondence to Strashimir G. Popvassilev.

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Popvassilev, S.G. (m,n)-Equidistant Sets in \(\mathbb{R}^{k},\mathbb{S}^{k}\) , and ℙk . Discrete Comput Geom 40, 279–288 (2008). https://doi.org/10.1007/s00454-007-9048-4

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Keywords

  • Projective Space
  • Unique Point
  • Great Circle
  • Equilateral Triangle
  • Discrete Comput Geom