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Radon partitions and a new notion of independence in affine and projective spaces

Published online by Cambridge University Press:  26 February 2010

J.-P. Doignon  and
G. Valette
J.-P. Doignon
Affiliation:
Départment de Mathématique (C.P.216), Université Libre de Bruxelles, Boulevard du Triomphe, B 1050 Bruxelles.
G. Valette
Affiliation:
Departement voor Wiskunde, Vrije Universiteit Brussel,Pleinlann 2, B 1050 Brussel
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Extract

It is well known that every set of at least d + 2 points of ℝd may be decomposed into two disjoint parts whose convex hulls intersect. This result, called Radon's theorem, has been generalized in different ways. The easiest generalization consists in replacing the field of real numbers by any ordered division ring (field or skew-field): here the original proof remains valid. A less immediate generalization is the following one, conjectured by Birch [1] and proved by Tverberg [3]:

Every set of at least r(d + 1) – d points of ℝd may be decomposed into r disjoint parts whose convex hulls intersect.

Type
Research Article
Information
Mathematika , Volume 24 , Issue 1 , June 1977 , pp. 86 - 96
Copyright
Copyright © University College London 1977

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References

1.Birch, B. J.. “On 3N points in a plane ”, Proc. Cambridge Phil. Soc, 55 (1959), 289–293.CrossRefGoogle Scholar
2.Reay, J. R.. “An extension of Radon's theorem ”, Illinois J. Math., 12 (1968), 184–189.CrossRefGoogle Scholar
3.Tverberg, H.. “A generalization of Radon's theorem ”, J. London Math. Soc, 41 (1966), 123–128.CrossRefGoogle Scholar