A counterexample to a “theorem” on $L_{n}$ sets
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- by John D. Baildon PDF
- Proc. Amer. Math. Soc. 73 (1979), 92-94 Request permission
Abstract:
An example is given of a closed connected set in $E^r$ whose points of local nonconvexity can be decomposed into two convex subsets, but which is not arcwise connected and hence is not an $L_n$ set. This contradicts a result by Valentine to which Stavrakas and Jamison have given a second proof. It is also shown that if the set of points of local nonconvexity of a closed connected set S in $E^r$ can be decomposed into n compact subsets which are convex relative to S, then S is an $L_{2n+1}$ set.References
- Nick M. Stavrakas and R. E. Jamison, Valentine’s extensions of Tietze’s theorem on convex sets, Proc. Amer. Math. Soc. 36 (1972), 229–230. MR 310763, DOI 10.1090/S0002-9939-1972-0310763-4
- F. A. Valentine, Local convexity and $L_{n}$ sets, Proc. Amer. Math. Soc. 16 (1965), 1305–1310. MR 185510, DOI 10.1090/S0002-9939-1965-0185510-6
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 73 (1979), 92-94
- MSC: Primary 52A20
- DOI: https://doi.org/10.1090/S0002-9939-1979-0512065-6
- MathSciNet review: 512065