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No Krasnoselskii Number for General Sets

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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.)

BibTeX - Entry

@InProceedings{keller_et_al:LIPIcs.SoCG.2021.47,
  author =	{Keller, Chaya and Perles, Micha A.},
  title =	{{No Krasnoselskii Number for General Sets}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{47:1--47:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13846},
  URN =		{urn:nbn:de:0030-drops-138462},
  doi =		{10.4230/LIPIcs.SoCG.2021.47},
  annote =	{Keywords: visibility, Helly-type theorems, Krasnoselskii’s theorem, transfinite induction, well-ordering theorem}
}

Keywords: visibility, Helly-type theorems, Krasnoselskii’s theorem, transfinite induction, well-ordering theorem
Seminar: 37th International Symposium on Computational Geometry (SoCG 2021)
Issue date: 2021
Date of publication: 02.06.2021


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