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Valentine convexity in n dimensions

Published online by Cambridge University Press:  24 October 2008

H. G. Eggleston
H. G. Eggleston
Affiliation:
Royal Holloway College, University of London
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Extract

A subset X of Euclidean space such that if a, b, c are points of X then at least one of the segments joining two of them lies in X, is said to be V-convex. Valentine (4) showed that in two dimensions a compact V-convex set is the union of at most three convex sets. We show here that if the set of star centres of X is of lower dimension than X and X is a compact V-convex set then it is the union of at most two convex sets.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 80 , Issue 2 , September 1976 , pp. 223 - 228
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

REFERENCES

(1)Descartes, S., Blanche A three colour problem. Eureka 9 (04 1947), 21. Solution. Eureka 10 (March 1948). See also solution to Advanced problem 1526. Amer. Math. Monthly 61 (1954), 352.Google Scholar
(2)Kelly, J. B. and Kelly, L. M.Paths and circuits in critical graphs. Amer. J. Math. 76 (1954), 786–792.CrossRefGoogle Scholar
(3)Mycelski, J.Sur le coloriage des graphs. Colloquium Mathematicum 3 (1955), 161–2.CrossRefGoogle Scholar
(4)Valentine, F. A.A three point convexity property. Pacific J. Math. 7 (1957), 1227–1235.CrossRefGoogle Scholar
(5)Zykov, A. A.On some properties of linear complexes. Math. Sbornik 66 (24) (1949), 163–188. Amer. Math. Soc. translation no. 79 (1952).Google Scholar