Abstract
Let S be a subset of R d. The set S is said to be an ℒ set if and only if for every two points x and y of S, there exists some z ∈ S such that [x, z] ⋃ [z, y] ⊂ S. Clearly every starshaped set is an ℒ set, yet the converse is false and introduces an interesting question: ‘Under what conditions will an ℒ set S be “almost” starshaped; that is, when will there exist a convex subset C of S such that every point of S sees some point of C via S’
This paper provides one answer to the question above, and we have the following result: Let S be a closed planar ℒ set, S simply connected, and assume that the set Q of points of local nonconvexity of S is finite. If some point p of S see each member of Q via S, then there is a convex subset C of S such that every point of S sees some point of C via S.
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Breen, M. ℒ2 sets which are almost starshaped. Geom Dedicata 6, 485–494 (1977). https://doi.org/10.1007/BF00147785
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DOI: https://doi.org/10.1007/BF00147785