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No Krasnoselskii Number for General Sets

Authors Chaya Keller, Micha A. Perles



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Author Details

Chaya Keller
  • Department of Computer Science, Ariel University, Israel
Micha A. Perles
  • Einstein Institute of Mathematics, Hebrew University, Jerusalem, Israel

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Chaya Keller and Micha A. Perles. No Krasnoselskii Number for General Sets. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 47:1-47:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.47

Abstract

For a family ℱ of non-empty sets in ℝ^d, 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^d. 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^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 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.)

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • visibility
  • Helly-type theorems
  • Krasnoselskii’s theorem
  • transfinite induction
  • well-ordering theorem

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