TY - JOUR

T1 - No Krasnosel’skiĭ number for general sets

AU - Keller, Chaya

AU - Perles, Micha A.

N1 - Publisher Copyright:
© 2023, The Hebrew University of Jerusalem.

PY - 2023/9

Y1 - 2023/9

N2 - For a family ℱ of non-empty sets in ℝd, the Krasnosel’skiĭ 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 Krasnosel’skiĭ number for general sets in ℝd.The best known positive result is Krasnosel’skiĭ number 3 for closed sets in the plane, and the best known negative result is that if a Krasnosel’skiĭ 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 Krasnosel’skiĭ 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 Krasnosel’skiĭ numbers with respect to visibility through polygonal paths of length ≤ n, for which an analogue of Krasnosel’skiĭ 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 Krasnosel’skiĭ 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 Krasnosel’skiĭ 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 Krasnosel’skiĭ number for general sets in ℝd.The best known positive result is Krasnosel’skiĭ number 3 for closed sets in the plane, and the best known negative result is that if a Krasnosel’skiĭ 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 Krasnosel’skiĭ 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 Krasnosel’skiĭ numbers with respect to visibility through polygonal paths of length ≤ n, for which an analogue of Krasnosel’skiĭ 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 Krasnosel’skiĭ 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.)

UR - http://www.scopus.com/inward/record.url?scp=85173710079&partnerID=8YFLogxK

U2 - 10.1007/s11856-023-2501-0

DO - 10.1007/s11856-023-2501-0

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AN - SCOPUS:85173710079

SN - 0021-2172

VL - 256

SP - 345

EP - 361

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -