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Ln Sets and the Closures of Open Connected Sets

Published online by Cambridge University Press:  20 November 2018

Nick M. Stavrakas
Nick M. Stavrakas*
Affiliation:
University of North Carolina at Charlotte, Charlotte, North Carolina
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F. A. Valentine in [4] proved the following two theorems.

THEOREM 1. Let S be a closed connected subset of Rd which has at most n points of local nonconvexity. Then S is an Ln+i set.

THEOREM 2. Let S be a closed connected subset of Rd whose points of local nonconvexity are decomposable into n closed convex sets. Then S is an L2n+i set.

These results have been extended by a number of authors, but always with stronger hypothesis. (See [1] and [2].) Using a minimal arc technique, new pr∞fs of Theorems 1 and 2 were given in [3].

Valentine remarks in [4] that Theorem 2 might be improved in the case that 5 is the closure of an open connected set. The goal of this paper is to give such an improvement for sets satisfying a particular local connectivity property.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 1 , 01 February 1975 , pp. 1 - 5
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Guay, M. and Kay, D., On sets having finitely many points of local nonconvexity and Property Pm, Israel J. Math. 10 (1971), 196–209.Google Scholar
2. Stavrakas, N. M., Hare, W. R., and Kenelly, J. W., Two cells with n points of local nonconvexity, Proc. Amer. Math. Soc. 27 (1971), 331–336.Google Scholar
3. Stavrakas, N. M. and Jamison, R. E., Valentine's extensions of Tietzës theorem on convex sets, Proc. Amer. Math. Soc. 86 (1972), 229–230.Google Scholar
4. Valentine, F. A., Local convexity and Ln sets, Proc. Amer. Math. Soc. 16 (1965), 1305–1310.Google Scholar