Summary
LetC be a compact set inR 2. A setS \( \subseteq \) R 2 ∼ C is said to have aj-partition relative toC if and only if there existj or fewer pointsc 1,⋯, c j inC such that each point ofS ‘sees somec i via the complement ofC’. Letm, j be fixed integers, 3 ⩽m, 2 ⩽j, and writem (uniquely) asm = qj + r, where 1 ⩽r ⩽j. Assume thatC is a convexm-gon in R2, withS \( \subseteq \) R 2 ∼ C. Forq = 0 orq = 1, the setS has aj-partition relative toC. Forq ⩾ 2,S has aj-partition relative toC if and only if every (qj + 1)-member subset ofS has aj-partition relative toC, and the Helly numberqj + 1 is best possible.
IfC is a disk, no such Helly number exists.
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References
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Breen, M. j-partitions for visible shorelines. Aeq. Math. 36, 201–211 (1988). https://doi.org/10.1007/BF01836091
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DOI: https://doi.org/10.1007/BF01836091