Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T00:28:48.005Z Has data issue: false hasContentIssue false
  • English
  • Français

The number of partitions of a set of N points in k dimensions induced by hyperplanes

Published online by Cambridge University Press:  20 January 2009

E. F. Harding
E. F. Harding
Affiliation:
Department of StatisticsUniversity of Edinburgh
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

1. An arbitrary (k– 1)-dimensional hyperplane disconnects K-dimensional Euclidean space Ek into two disjoint half-spaces. If a set of N points in general position in Ek is given [nok +1 in a (k–1)-plane, no k in a (k–2)-plane, and so on], then the set is partitione into two subsets by the hyperplane, a point belonging to one or the other subset according to which half-space it belongs to; for this purpose the half-spaces are considered as an unordered pair.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 15 , Issue 4 , December 1967 , pp. 285 - 289
Copyright
Copyright © Edinburgh Mathematical Society 1967

References

REFERENCES

(1) Schläfli, L., Theorie der vielfachen Kontinuitāt (Berne, 1852), Ges. Math. Abh. Vol.1, p. 209 (Basel, 1950).Google Scholar
(2) Carver, W. B., The polygonal regions into which a plane is divided by n straight lines. Amer. Math. Monthly, 48 (1941), 667–675.CrossRefGoogle Scholar
(3) Blumenthal, L. M. and Gillam, B. E, Distribution of points in n-space. Amer. Math. Monthly, 50 (1943), 181–185.CrossRefGoogle Scholar
(4) Buck, R. C., Partition of space, Amer. Math. Monthly, 50 (1943), 541–544.CrossRefGoogle Scholar
(5) Robinson, H. A., A problem of regions. Amer. Math. Monthly 52 (1945), 34.CrossRefGoogle Scholar
(6) Edwards, A. W. F. and Cavalli-Sforza, L. L., A method for Cluster Analysis, Biometrics, 21 (2) (1965), 362–375.CrossRefGoogle ScholarPubMed
(7) Winder, R. O., Partitions of N-space by Hyperplanes, SIAM J. Appl. Math. 14 (4) (1966), 811–818.CrossRefGoogle Scholar