2026-08-01T13:22:59Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:13846 2024-03-06T10:52:52Z ddc:004 open_access

No Krasnoselskii Number for General Sets Keller, Chaya Perles, Micha A. visibility Helly-type theorems Krasnoselskii’s theorem transfinite induction well-ordering theorem 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.) Schloss Dagstuhl – Leibniz-Zentrum für Informatik Chaya Keller and Micha A. Perles 2021 Is Part Of LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2021.47 urn:nbn:de:0030-drops-138462 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.47 eng https://creativecommons.org/licenses/by/4.0/legalcode