Abstract
If a pointq ofS has the property that each neighborhood ofq contains pointsx andy such that the segmentxy is not contained byS, q is called a point of local nonconvexity ofS. LetQ denote the set of points of local nonconvexity ofS. Tietze’s well known theorem that a closed connected setS in a linear topological space is convex ifQ=φ is generalized in the result:If S is a closed set in a linear topological space such that S ∼ Q is connected and |Q|=n<∞,then S is the union of n+1or fewer closed convex sets. Letk be the minimal number of convex sets needed in a convex covering ofS. Bounds fork in terms ofm andn are obtained for sets having propertyP m and |Q|=n.
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Guay, M.D., Kay, D.C. On sets having finitely many points of local nonconvexity and propertyP m . Israel J. Math. 10, 196–209 (1971). https://doi.org/10.1007/BF02771570
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DOI: https://doi.org/10.1007/BF02771570