Geodesic
Ln Sets and the Closures of Open Connected Sets
Canadian journal of mathematics, Tome 27 (1975) no. 1, pp. 1-5

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Stavrakas, Nick M. Ln Sets and the Closures of Open Connected Sets. Canadian journal of mathematics, Tome 27 (1975) no. 1, pp. 1-5. doi: 10.4153/CJM-1975-001-3
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[1] 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] 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] 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] 4. Valentine, F. A., Local convexity and Ln sets, Proc. Amer. Math. Soc. 16 (1965), 1305–1310. Google Scholar

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