An (?0, k + 2)-Theorem for k-Transversals

Chaya Keller, Micha A. Perles

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

Abstract

A family F of sets satisfies the (p, q)-property if among every p members of F, some q can be pierced by a single point. The celebrated (p, q)-theorem of Alon and Kleitman asserts that for any p ? q ? d + 1, any family F of compact convex sets in Rd that satisfies the (p, q)-property can be pierced by a finite number c(p, q, d) of points. A similar theorem with respect to piercing by (d - 1)-dimensional flats, called (d - 1)-transversals, was obtained by Alon and Kalai. In this paper we prove the following result, which can be viewed as an (?0, k + 2)-theorem with respect to k-transversals: Let F be an infinite family of sets in Rd such that each A ? F contains a ball of radius r and is contained in a ball of radius R, and let 0 ? k < d. If among every ?0 elements of F, some k + 2 can be pierced by a k-dimensional flat, then F can be pierced by a finite number of k-dimensional flats. This is the first (p, q)-theorem in which the assumption is weakened to an (8, ·) assumption. Our proofs combine geometric and topological tools.

Original languageEnglish
Title of host publication38th International Symposium on Computational Geometry, SoCG 2022
EditorsXavier Goaoc, Michael Kerber
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772273
DOIs
StatePublished - 1 Jun 2022
Externally publishedYes
Event38th International Symposium on Computational Geometry, SoCG 2022 - Berlin, Germany
Duration: 7 Jun 202210 Jun 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume224
ISSN (Print)1868-8969

Conference

Conference38th International Symposium on Computational Geometry, SoCG 2022
Country/TerritoryGermany
CityBerlin
Period7/06/2210/06/22

Keywords

  • (p,q)-theorem
  • convexity
  • infinite (p,q)-theorem
  • k-transversal

ASJC Scopus subject areas

  • Software

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