No Krasnoselskii Number for General Sets

Abstract

For a family F of non-empty sets in R, the Krasnoselskii number of F is the smallest m such that for any S ∈ F , if every m or fewer 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. The best known positive result is Krasnoselskii number 3 for closed sets in the plane, and the best known negative result is that if a Krasnoselskii number for general sets in R exists, it cannot be smaller than (d + 1)2. 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 R2. 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 for compact simply connected sets 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 compact sets in R2 with respect to visibility through paths of length ≤ n. (Here the counterexamples are finite unions of line segments.) 2012 ACM Subject Classification Theory of computation → Computational geometry