Summary. Let C be a compact set in R z. A set S ~_ R 2 ~ C is said to have a j-partition relative to C if and only if there exist j or fewer points c I .... , cj in C such that each point of S 'sees some c i via the complement of C'. Let m,j be fixed integers, 3 ~< m, 2 ~< j, and write m (uniquely) as m = qj + r, where 1 ~< r ~< j. Assume that C is a convex m-gon in R 2, with S ~_ R 2 ~ C. For q = 0 or q = 1, the set S has a j-partition relative to C. For q t> 2, S has a j-partition relative to C if and only if every (qj + 1)-member subset of S has a j-partition relative to C, and the Helly number qj + 1 is best possible.If C is a disk, no such Helly number exists.
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