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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.

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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.

References

  1. Noga Alon and Gil Kalai. Bounding the piercing number. Discrete & Computational Geometry, 13:245-256, 1995. URL: https://doi.org/10.1007/BF02574042.
  2. Noga Alon, Gil Kalai, Jiří Matoušek, and Roy Meshulam. Transversal numbers for hypergraphs arising in geometry. Advances in Applied Mathematics, 29(1):79-101, 2002. URL: https://doi.org/10.1016/S0196-8858(02)00003-9.
  3. Noga Alon and Daniel J Kleitman. Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem. Advances in Mathematics, 96(1):103-112, 1992. Google Scholar
  4. Boris Aronov, Jacob E Goodman, and Richard Pollack. A Helly-type theorem for higher-dimensional transversals. Computational Geometry, 21(3):177-183, 2002. URL: https://doi.org/10.1016/S0925-7721(01)00025-6.
  5. Imre Bárány, Ferenc Fodor, Luis Montejano, Deborah Oliveros, and Attila Pór. Colourful and fractional (p, q)-theorems. Discrete & Computational Geometry, 51(3):628-642, 2014. URL: https://doi.org/10.1007/S00454-013-9559-0.
  6. Arijit Ghosh and Soumi Nandi. Heterochromatic higher order transversals for convex sets. arXiv preprint arXiv:2212.14091 version 1, 2022. URL: https://doi.org/10.48550/arXiv.2212.14091.
  7. Jacob E Goodman, Richard Pollack, and Rephael Wenger. Geometric transversal theory. In New trends in discrete and computational geometry, pages 163-198. Springer, 1993. Google Scholar
  8. Hugo Hadwiger and Hans Debrunner. Über eine Variante zum Hellyschen Satz. Archiv der Mathematik, 8:309-313, 1957. URL: https://doi.org/10.1007/BF01898794.
  9. David Haussler and Emo Welzl. Epsilon-nets and simplex range queries. In Proceedings of the second annual symposium on Computational geometry, pages 61-71, 1986. URL: https://doi.org/10.1145/10515.10522.
  10. Eduard Helly. Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahresbericht der Deutschen Mathematiker-Vereinigung, 32:175-176, 1923. Google Scholar
  11. Andreas Holmsen and Jiří Matoušek. No Helly theorem for stabbing translates by lines in ℝ³. Discrete & Computational Geometry, 31(3):405-410, 2004. URL: https://doi.org/10.1007/s00454-003-0796-5.
  12. Fritz John. Extremum problems with inequalities as subsidiary conditions. studies and essays presented to R. Courant on his 60th birthday, 1948. Google Scholar
  13. Attila Jung and Dömötör Pálvölgyi. A note on infinite versions of (p, q)-theorems. arXiv preprint, 2024. URL: https://arxiv.org/abs/2412.04066.
  14. Meir Katchalski and Andy Liu. A problem of geometry in Rⁿ. Proceedings of the American Mathematical Society, 75(2):284-288, 1979. URL: https://doi.org/10.2307/2042758.
  15. Chaya Keller and Micha A. Perles. An (ℵ₀, k+ 2)-theorem for k-transversals. 38th International Symposium on Computational Geometry (SoCG 2022) and arXiv preprint arXiv:2306.02181, 2022-2023. URL: https://arxiv.org/abs/2306.02181.
  16. Jiří Matoušek. Bounded VC-dimension implies a fractional Helly theorem. Discrete & Computational Geometry, 31:251-255, 2004. URL: https://doi.org/10.1007/S00454-003-2859-Z.
  17. Daniel McGinnis and Nikola Sadovek. A necessary and sufficient condition for k-transversals. arXiv preprint, 2024. URL: https://arxiv.org/abs/2411.07241.
  18. John Milnor. On the Betti numbers of real varieties. Proceedings of the American Mathematical Society, 15(2):275-280, 1964. Google Scholar
  19. Luis A. Santaló. A theorem on sets of parallelepipeds with parallel edges. Publicaciones del Instituto de Matemática de la Universidad Nacional del Litoral, 2:49-60, 1940. Google Scholar
  20. René Thom. Sur l’homologie des variétés algébriques réelles. Differential and combinatorial topology, pages 255-265, 1965. Google Scholar
  21. Paul Vincensini. Figures convexes et variétés linéaires de l’espace euclidiena n dimensions. Bulletin des Sciences Mathématiques, 59:163-174, 1935. Google Scholar
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