TY - GEN

T1 - No Krasnoselskii number for general sets

AU - Keller, Chaya

AU - Perles, Micha A.

N1 - Publisher Copyright:
© Chaya Keller and Micha A. Perles; licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021).

PY - 2021/6/1

Y1 - 2021/6/1

N2 - 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).

AB - 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).

KW - Helly-type theorems

KW - Krasnoselskii's theorem

KW - Transfinite induction

KW - Visibility

KW - Well-ordering theorem

UR - http://www.scopus.com/inward/record.url?scp=85108191107&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2021.47

DO - 10.4230/LIPIcs.SoCG.2021.47

M3 - Conference contribution

AN - SCOPUS:85108191107

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 37th International Symposium on Computational Geometry, SoCG 2021

A2 - Buchin, Kevin

A2 - de Verdiere, Eric Colin

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 37th International Symposium on Computational Geometry, SoCG 2021

Y2 - 7 June 2021 through 11 June 2021

ER -