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 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 of the points. Thus in euclidean space we prove that if the cofactor of the element in the determinant is nonzero then lie in the same or different components of (where denotes the sphere or hyperplane containing the remaining points) if and only if or , respectively. In hyperbolic space the result is: if the cofactor of the element in the determinant is nonzero then lie in the same or different components of (where denotes the hyperplane, sphere, horosphere, or one branch of an equidistant surface containing the remaining points) if and only if or , respectively. For spherical space we obtain: if the cofactor of the element in the determinant is nonzero then lie in the same or different components of (where denotes the sphere containing the remaining points which may be an dimensional subspace) if and only if or respectively.

**[1]**L. M. Blumenthal,*The geometry of a class of semimetric spaces*, Tôhoku Math. J.**43**(1937), 205-224.**[2]**Leonard M. Blumenthal,*The metric characterization of 𝜙-spherical spaces*, Univ. Nac. Tucumán. Revista A.**5**(1946), 69–93 (Spanish). MR**0020779****[3]**L. M. Blumenthal and B. E. Gillam,*Distribution of points in 𝑛-space*, Amer. Math. Monthly**50**(1943), 181–185. MR**0008144**, https://doi.org/10.2307/2302400**[4]**A. Cayley,*On a theorem in the geometry of position*, Cambridge Math. J. 2 (1841), 267-271.**[5]**Gaston Darboux,*Sur les relations entre les groupes de points, de cercles et de sphères dans le plan et dans l’espace*, Ann. Sci. École Norm. Sup. (2)**1**(1872), 323–392 (French). MR**1508589****[6]**Thomas Muir,*A treatise on the theory of determinants*, Revised and enlarged by William H. Metzler, Dover Publications, Inc., New York, 1960. MR**0114826****[7]**K. Menger,*Bericht über metrische geometric*, Jber. Deutsch. Math. Verein.**40**(1931), 201-219.**[8]**D. M. Y. Sommerville,*The elements of non-Euclidean geometry*, G. Bell, London, 1914.**[9]**J. E. Valentine,*An analogue of Ptolemy’s theorem in spherical geometry*, Amer. Math. Monthly**77**(1970), 47–51. MR**0254725**, https://doi.org/10.2307/2316853**[10]**Joseph E. Valentine,*An analogue of Ptolemy’s theorem and its converse in hyperbolic geometry*, Pacific J. Math.**34**(1970), 817–825. MR**0270261****[11]**Joseph E. Valentine and Edward Z. Andalafte,*A metric characterization of “spherical” surfaces in 𝑛-dimensional hyperbolic space*, J. Reine Angew. Math.**251**(1971), 142–152. MR**0293481**, https://doi.org/10.1515/crll.1971.251.142

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Additional Information

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