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A proof of a theorem of Valentine

Published online by Cambridge University Press:  24 October 2008

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

An interesting extension of the idea of convexity has been introduced by Valentine (2). He considered a plane set X such that for any three points a, b, c in X at least one of the segments [a, b], [b, c], [c, a] is contained in X, and showed that such a set can be regarded as the union of at most three convex sets. See also (3). The five pointed star (Fig. 1) is an example that shows that X may not be the union of fewer than three convex sets.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 77 , Issue 3 , May 1975 , pp. 525 - 528
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

REFERENCES

(1)Eggleston, H. G.Approximation to plane convex curves (1) Dowker-type theorems. Proc. London Math. Soc. (3) 7 (1957), 351–377.CrossRefGoogle Scholar
(2)Valentine, F. A.A three point convexity property. Pacific J. Math. 7 (1957), 1227–1235. (See also Combinatorial Geometry in the plane, Hadwiger, Debrunner and Klee Ex. 51 p. 72.)CrossRefGoogle Scholar
(3)Stamey, W. L. and Marr, J. M.Unions of two convex sets. Can. J. Math. 15 (1963), 152–6.CrossRefGoogle Scholar