No Krasnoselskii number for general sets in ℝ^2
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For a family ℱ of sets in ℝ^d, the Krasnoselskii number of ℱ is the smallest m such that for any S ∈ℱ, if every m points of S are visible from a common point in S, then any finite subset of S is visible from a single point. More than 35 years ago, Peterson asked whether there exists a Krasnoselskii number for general sets in ℝ^d. Excluding results for special cases of sets with strong topological restrictions, the best known result is due to Breen, who showed that if such a Krasnoselskii number in ℝ^2 exists, then it is larger than 8. In this paper we answer Peterson's question in the negative by showing that there is no Krasnoselskii number for the family of all sets in ℝ^2. The proof is non-constructive, and uses transfinite induction and the well ordering theorem. In addition, we consider Krasnoselskii numbers with respect to visibility through polygonal paths of length ≤ n, for which an analogue of Krasnoselskii's theorem was proved by Magazanik and Perles. We show, by an explicit construction, that for any n ≥ 2, there is no Krasnoselskii number for the family of general sets in ℝ^2 with respect to visibility through paths of length ≤ n. (Here the counterexamples are finite unions of line segments.)
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