Abstract
LetS be a compact, simply connected set inR 2. If every boundary point ofS is clearly visible viaS from at least one of the three pointsa, b, c, thenS is a union of three starshaped sets whose kernels containa, b, c, respectively. The result fails when the number three is replaced by four.
As a partial converse, ifS is a union of three starshaped sets whose kernels containa, b, c, respectively, then the set of points in the boundary ofS clearly visible from at least one ofa, b, orc is dense in the boundary ofS.
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References
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Supported in part by NSF grant DMS-8705336.
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Breen, M. Unions of three starshaped sets in R2 . J Geom 36, 8–16 (1989). https://doi.org/10.1007/BF01231019
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DOI: https://doi.org/10.1007/BF01231019