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Proceedings of the American Mathematical Society

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A counterexample to a ``theorem'' on $ L\sb{n}$ sets


Author: John D. Baildon
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
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1979-0512065-6
Keywords: Convex, locally convex, local nonconvexity, $ {L_n}$ set, Tietze's theorem
Article copyright: © Copyright 1979 American Mathematical Society