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Transactions of the American Mathematical Society

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A necessary and sufficient condition for a ``sphere'' to separate points in euclidean, hyperbolic, or spherical space


Authors: J. E. Valentine and S. G. Wayment
Journal: Trans. Amer. Math. Soc. 176 (1973), 285-295
MSC: Primary 50C05; Secondary 50B10
DOI: https://doi.org/10.1090/S0002-9947-1973-0313927-4
MathSciNet review: 0313927
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Abstract: The purpose of this paper is to give conditions wholly and explicitly in terms of the mutual distances of $ n + 3$ points in n-space which are necessary and sufficient for two of the points to lie in the same or different components of the space determined by the sphere which is determined by $ n + 1$ of the points. Thus in euclidean space we prove that if the cofactor $ [{p_i}{p_j}^2]$ of the element $ {p_i}{p_j}^2\;(i \ne j)$ in the determinant $ \vert{p_i}{p_j}^2\vert(i,j = 0,1, \cdots ,n + 2)$ is nonzero then $ {p_i},{p_j}$ lie in the same or different components of $ {E_n} - \Omega $ (where $ \Omega $ denotes the sphere or hyperplane containing the remaining $ n + 1$ points) if and only if $ \operatorname{sgn} [{p_i}{p_j}^2] = {( - 1)^n}$ or $ {( - 1)^{n + 1}}$, respectively. In hyperbolic space the result is: if the cofactor $ [{\sinh ^2}\;{p_i}{p_j}/2]$ of the element $ {\sinh ^2}\;{p_i}{p_j}/2\;(i \ne j)$ in the determinant $ \vert{\sinh ^2}\;{p_i}{p_j}/2\vert(i,j = 0,1, \cdots ,n + 1)$ is nonzero then $ {p_i},{p_j}$ lie in the same or different components of $ {H_n} - \Omega $ (where $ \Omega $ denotes the hyperplane, sphere, horosphere, or one branch of an equidistant surface containing the remaining $ n + 1$ points) if and only if $ \operatorname{sgn} [{\sinh ^2}\;{p_i}{p_j}/2] = {( - 1)^n}$ or $ {( - 1)^{n + 1}}$, respectively. For spherical space we obtain: if the cofactor $ [{\sin ^2}\;{p_i}{p_j}/2]$ of the element $ {\sin ^2}\;{p_i}{p_j}/2\;(i \ne j)$ in the determinant $ \vert{\sin ^2}\;{p_i}{p_j}/2\vert(i,j = 0,1, \cdots ,n + 2)$ is nonzero then $ {p_i},{p_j}$ lie in the same or different components of $ {S_n} - \Omega $ (where $ \Omega $ denotes the sphere containing the remaining $ n + 1$ points which may be an $ (n - 1)$ dimensional subspace) if and only if $ \operatorname{sgn} [{\sin ^2}\;{p_i}{p_j}/2] = {( - 1)^n}$ or $ {( - 1)^{n + 1}}$ respectively.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0313927-4
Keywords: Components, determinant, euclidean, hyperbolic, sphere, spherical
Article copyright: © Copyright 1973 American Mathematical Society