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

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.