# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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## A necessary and sufficient condition for a “sphere” to separate points in euclidean, hyperbolic, or spherical spaceHTML articles powered by AMS MathViewer

by J. E. Valentine and S. G. Wayment
Trans. Amer. Math. Soc. 176 (1973), 285-295 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 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 $|{p_i}{p_j}^2|(i,j = 0,1, \cdots ,n + 2)$ 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.
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