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A characterization of sets of n points which determine n hyperplanes†

Published online by Cambridge University Press:  24 October 2008

J. G. Basterfield  and
L. M. Kelly
J. G. Basterfield
Affiliation:
Emmanuel College, Cambridge
L. M. Kelly
Affiliation:
Michigan State University, East Lansing, Michigan
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Extract

Suppose N is a set of points of a d-dimensional incidence space S and {Ha}, a ∈ I, a set of hyperplanes of S such that Hi ∈ {Ha} if and only if Hi ∩ N spans Hi. N is then said to determine {Ha}. We are interested here in the case in which N is a finite set of n points in S and I = {1, 2,…, n}; that is to say when a set of n points determines precisely n hyperplanes. Such a situation occurs in E3, for example, when N spans E3 and is a subset of two (skew) lines, or in E2 if N spans the space and n − 1 of the points are on a line. On the other hand, the n points of a finite projective space determine precisely n hyperplanes so that the structure of a set of n points determining n hyperplanes is not at once transparent.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 3 , July 1968 , pp. 585 - 588
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

REFERENCES

(1)Birkhoff, Garrett. Lattice theory. Amer. Math. Soc. Coll. Publ. vol. xxv, 1948.Google Scholar
(2)Bruijn, N. G. de and Erdös, P.On a combinatorial problem. K. Nederl. Acad. Wetensch. Proc. Ser. A. 51 (1948), 1277–1279.Google Scholar
(3)Hanani, H.On the number of lines and planes determined by d points. Tech. Sci. Publ. Vol. 6, 1945/1946.Google Scholar
(4)Kelly, L. M. and Moser, W. O. J.On the number of ordinary lines determined by n points. Canad. J. Math. 10 (1958), 210–219.CrossRefGoogle Scholar
(5)Motzkin, Th.The lines and planes connecting the points of a finite set. Trans. Amer. Math. Soc. 70 (1951), 451–464.CrossRefGoogle Scholar