No Krasnoselskii number for general sets

Chaya Keller, Micha A. Perles

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

Abstract

For a family F of non-empty sets in ℝd, 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 ℝd. 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 ℝd 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).

Original languageEnglish
Title of host publication37th International Symposium on Computational Geometry, SoCG 2021
EditorsKevin Buchin, Eric Colin de Verdiere
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771849
DOIs
StatePublished - 1 Jun 2021
Externally publishedYes
Event37th International Symposium on Computational Geometry, SoCG 2021 - Virtual, Buffalo, United States
Duration: 7 Jun 202111 Jun 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume189
ISSN (Print)1868-8969

Conference

Conference37th International Symposium on Computational Geometry, SoCG 2021
Country/TerritoryUnited States
CityVirtual, Buffalo
Period7/06/2111/06/21

Keywords

  • Helly-type theorems
  • Krasnoselskii's theorem
  • Transfinite induction
  • Visibility
  • Well-ordering theorem

ASJC Scopus subject areas

  • Software

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