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k-Dimensional Transversals for Fat Convex Sets

Authors Attila Jung , Dömötör Pálvölgyi

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ACM Subject Classification
  • Mathematics of computing → Hypergraphs
  • Theory of computation → Computational geometry
Keywords
  • discrete geometry
  • transversals
  • Helly
  • hypergraphs

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Abstract

We prove a fractional Helly theorem for k-flats intersecting fat convex sets. A family ℱ of sets is said to be ρ-fat if every set in the family contains a ball and is contained in a ball such that the ratio of the radii of these balls is bounded by ρ. We prove that for every dimension d and positive reals ρ and α there exists a positive β = β(d,ρ, α) such that if ℱ is a finite family of ρ-fat convex sets in ℝ^d and an α-fraction of the (k+2)-size subfamilies from ℱ can be hit by a k-flat, then there is a k-flat that intersects at least a β-fraction of the sets of ℱ. We prove spherical and colorful variants of the above results and prove a (p,k+2)-theorem for k-flats intersecting balls.

Cite As Get BibTex

Attila Jung and Dömötör Pálvölgyi. k-Dimensional Transversals for Fat Convex Sets. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 61:1-61:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.61

Author Details

Attila Jung
  • ELTE Eötvös Loránd University, Budapest, Hungary
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Dömötör Pálvölgyi
  • ELTE Eötvös Loránd University, Budapest, Hungary
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary

Funding

  • Jung, Attila: Supported by the ERC Advanced Grant "ERMiD", by the EXCELLENCE-24 project no. 151504 Combinatorics and Geometry of the NRDI Fund, and by the NKFIH grants TKP2021-NKTA-62, FK132060 and SNN135643.
  • Pálvölgyi, Dömötör: Supported by the ERC Advanced Grant "ERMiD", by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the grants TKP2021-NKTA-62, ÚNKP-23-5 and EXCELLENCE-24 project no. 151504 Combinatorics and Geometry of the NRDI Fund.

Acknowledgements

We would like to thank Andreas Holmsen for useful discussions, and for bringing [Matoušek, 2004] to our attention.

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