Abstract
For a family ℱ of nonempty 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 nonconstructive, and uses transfinite induction and the wellordering 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 = {47:147:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771849},
ISSN = {18688969},
year = {2021},
volume = {189},
editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13846},
URN = {urn:nbn:de:0030drops138462},
doi = {10.4230/LIPIcs.SoCG.2021.47},
annote = {Keywords: visibility, Hellytype theorems, Krasnoselskii’s theorem, transfinite induction, wellordering theorem}
}
Keywords: 

visibility, Hellytype theorems, Krasnoselskii’s theorem, transfinite induction, wellordering theorem 
Collection: 

37th International Symposium on Computational Geometry (SoCG 2021) 
Issue Date: 

2021 
Date of publication: 

02.06.2021 