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The graph-theoretic analogue of Tietze's characterization of a convex set and its generalization†

Published online by Cambridge University Press:  24 October 2008

M. D. Guay
M. D. Guay
Affiliation:
University of Maine
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Extract

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 1 , July 1972 , pp. 7 - 9
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Harary, F.Graph theory (Addison-Wesley; Reading, Massachusetts, 1969).Google Scholar
(2)Kay, D. C. and Guay, M. D. On sets having finitely many points of local nonconvexity and property P m.Israel J. Math. (to appear).Google Scholar
(3)Valentine, F. A.Convex sets (McGraw-Hill; New York, 1964).Google Scholar