No Krasnosel’skiĭ number for general sets

Abstract

For a family ${\cal F}$ of non-empty sets in ℝ^d, the Krasnosel’skiĭ number of ${\cal F}$ is the smallest m such that for any $S \in {\cal F}$, 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 Krasnosel’skiĭ number for general sets in ℝ^d.The best known positive result is Krasnosel’skiĭ number 3 for closed sets in the plane, and the best known negative result is that if a Krasnosel’skiĭ number for general sets in ℝ^d 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 Krasnosel’skiĭ 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 Krasnosel’skiĭ numbers with respect to visibility through polygonal paths of length ≤ n, for which an analogue of Krasnosel’skiĭ 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 Krasnosel’skiĭ 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.)