A necessary and sufficient condition for a “sphere” to separate points in euclidean, hyperbolic, or spherical space
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 by J. E. Valentine and S. G. Wayment PDF
 Trans. Amer. Math. Soc. 176 (1973), 285295 Request permission
Abstract:
The purpose of this paper is to give conditions wholly and explicitly in terms of the mutual distances of $n + 3$ points in nspace 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 ${p_i}{p_j}^2(i,j = 0,1, \cdots ,n + 2)$ 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 ${\sinh ^2}\;{p_i}{p_j}/2(i,j = 0,1, \cdots ,n + 1)$ 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 ${\sin ^2}\;{p_i}{p_j}/2(i,j = 0,1, \cdots ,n + 2)$ 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.References

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Additional Information
 © Copyright 1973 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 176 (1973), 285295
 MSC: Primary 50C05; Secondary 50B10
 DOI: https://doi.org/10.1090/S00029947197303139274
 MathSciNet review: 0313927