Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T07:55:47.585Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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