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A condition for a compact plane set to be a union of finitely many convex sets

Published online by Cambridge University Press:  24 October 2008

H. G. Eggleston
H. G. Eggleston
Affiliation:
Royal Holloway College
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Extract

All the sets with which we are concerned are subsets of the real Euclidean plane E2. By Lm we denote those subsets X of E2 for which, if pl, p2, …, Pm are any m points of X, then at least one segment pipj, i ≠ j consists entirely of points of X. L2 is the class of convex subsets of E2. We shall show that if X is closed and X ∈ Lm. then X is the union of finitely many convex sets. This extends a result of Valentine (4). See also (1),(2),(3).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 1 , July 1974 , pp. 61 - 66
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

(1)Buchan, E. O.Property P 3 and the union of two convex sets. Proc. Amer. Math. Soc. 25 (1970), 642–645.Google Scholar
(2)Hare, W. R. Jr, and Kenelly, John W.Sets expressible as unions of two convex sets. Proc. Amer. Math. Soc. 25 (1970), 379–380.CrossRefGoogle Scholar
(3)McKinsey, Richard L.On unions of two convex sets. Can. J. Math. 18 (1966), 883–886.Google Scholar
(4)Valentine, F. A.A three point convexity property. Pacific J. Math. 7 (1957), 1227–1235, M.R. 20, no. 6071. See also Hadwiger, Debrunner and Klee, Combinatorial geometry in the plane (Example 51, p. 72) (Holt, 1964).Google Scholar