Abstract
The following Krasnosel'skii-type theorem is proved: Let S be a nonempty set in R2 whose closure cl S is convex and bounded. Assume that for every 9 point subset T of cl S there correspond points p1 and p2 (depending on T) such that each point of T is clearly visible via S from at least one of p1 or p2. Then S is a union of two starshaped sets. The number 9 is best possible.
Moreover, a related result yields a piercing number for families of segments in Rd: Let £ be a collection of at least 6 one-dimensional convex sets in Rd such that for every line M in Rd, at most finitely many members of £ are collinear with M. Assume that every 6 members of £ may be partitioned into two sets £1 and £2 so that ∩L ∶ L in £i ≠ φ, i = 1,2. Then £ itself has such a 2-partition. The number 6 is best possible as well.
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Dedicated to Professor M. Barner on his 65th birthday
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Breen, M. Sets with convex closure which are unions of two starshaped sets and families of segments which have a 2-partition. J Geom 27, 1–23 (1986). https://doi.org/10.1007/BF01230330
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DOI: https://doi.org/10.1007/BF01230330