Document Open Access Logo

Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers

Authors Elizaveta Streltsova , Uli Wagner

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.SoCG.2025.75.pdf
  • Filesize: 0.9 MB
  • 18 pages
HTML5 Logo HTML (experimental)
LIPIcs.SoCG.2025.75.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Discrete mathematics
Keywords
  • Levels in arrangements
  • k-sets
  • k-facets
  • convex polytopes
  • f-vector
  • h-vector
  • g-vector
  • Dehn-Sommerville relations
  • Radon partitions
  • Gale duality
  • g-matrix

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

A long-standing conjecture of Eckhoff, Linhart, and Welzl, which would generalize McMullen’s Upper Bound Theorem for polytopes and refine asymptotic bounds due to Clarkson, asserts that for k ⩽ ⌊(n-d-2)/2⌋, the complexity of the (⩽ k)-level in a simple arrangement of n hemispheres in S^d is maximized for arrangements that are polar duals of neighborly d-polytopes. We prove this conjecture in the case n = d+4. By Gale duality, this implies the following result about crossing numbers: In every spherical arc drawing of K_n in S² (given by a set V ⊂ S² of n unit vectors connected by spherical arcs), the number of crossings is at least 1/4 ⌊n/2⌋ ⌊(n-1)/2⌋ ⌊(n-2)/2⌋ ⌊(n-3)/2⌋. This lower bound is attained if every open linear halfspace contains at least ⌊(n-2)/2⌋ of the vectors in V.
Moreover, we determine the space of all linear and affine relations that hold between the face numbers of levels in simple arrangements of n hemispheres in S^d. This completes a long line of research on such relations, answers a question posed by Andrzejak and Welzl in 2003, and generalizes the classical fact that the Dehn-Sommerville relations generate all linear relations between the face numbers of simple polytopes (which correspond to the 0-level).
To prove these results, we introduce the notion of the g-matrix, which encodes the face numbers of levels in an arrangement and generalizes the classical g-vector of a polytope.

Cite As Get BibTex

Elizaveta Streltsova and Uli Wagner. Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 75:1-75:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.75

Author Details

Elizaveta Streltsova
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Uli Wagner
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria

References

  1. Bernardo M. Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Dan McQuillan, Bojan Mohar, Petra Mutzel, Pedro Ramos, R. Bruce Richter, and Birgit Vogtenhuber. Bishellable drawings of K_n. SIAM J. Discrete Math., 32(4):2482-2492, 2018. URL: https://doi.org/10.1137/17M1147974.
  2. Bernardo M. Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Pedro Ramos, and Gelasio Salazar. The 2-page crossing number of K_n. Discrete Comput. Geom., 49(4):747-777, 2013. URL: https://doi.org/10.1007/S00454-013-9514-0.
  3. Bernardo M. Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Pedro Ramos, and Gelasio Salazar. Shellable drawings and the cylindrical crossing number of K_n. Discrete Comput. Geom., 52(4):743-753, 2014. URL: https://doi.org/10.1007/S00454-014-9635-0.
  4. Bernardo M. Ábrego, József Balogh, Silvia Fernández-Merchant, Jesús Leaños, and Gelasio Salazar. An extended lower bound on the number of (⩽ k)-edges to generalized configurations of points and the pseudolinear crossing number of K_n. J. Combin. Theory Ser. A, 115(7):1257-1264, 2008. URL: https://doi.org/10.1016/J.JCTA.2007.11.004.
  5. Bernardo M. Ábrego, Mario Cetina, Silvia Fernández-Merchant, Jesús Leaños, and Gelasio Salazar. On ⩽ k-edges, crossings, and halving lines of geometric drawings of K_n. Discrete Comput. Geom., 48(1):192-215, 2012. URL: https://doi.org/10.1007/S00454-012-9403-Y.
  6. Bernardo M. Ábrego and Silvia Fernández-Merchant. A lower bound for the rectilinear crossing number. Graphs Combin., 21(3):293-300, 2005. URL: https://doi.org/10.1007/S00373-005-0612-5.
  7. Bernardo M. Ábrego, Silvia Fernández-Merchant, Jesús Leaños, and Gelasio Salazar. A central approach to bound the number of crossings in a generalized configuration. In The IV Latin-American Algorithms, Graphs, and Optimization Symposium, volume 30 of Electron. Notes Discrete Math., pages 273-278. Elsevier Sci. B. V., Amsterdam, 2008. URL: https://doi.org/10.1016/J.ENDM.2008.01.047.
  8. Bernardo M. Ábrego, Silvia Fernández-Merchant, and Gelasio Salazar. The rectilinear crossing number of K_n: closing in (or are we?). In Thirty essays on geometric graph theory, pages 5-18. Springer, New York, 2013. Google Scholar
  9. Oswin Aichholzer, Jesús García, David Orden, and Pedro Ramos. New lower bounds for the number of (⩽ k)-edges and the rectilinear crossing number of K_n. Discrete Comput. Geom., 38(1):1-14, 2007. URL: https://doi.org/10.1007/S00454-007-1325-8.
  10. Noga Alon and E. Győri. The number of small semispaces of a finite set of points in the plane. J. Combin. Theory Ser. A, 41(1):154-157, 1986. URL: https://doi.org/10.1016/0097-3165(86)90122-6.
  11. Artur Andrzejak, Boris Aronov, Sariel Har-Peled, Raimund Seidel, and Emo Welzl. Results on k-sets and j-facets via continuous motion. In Proceedings of the Fourteenth Annual Symposium on Computational Geometry, SCG '98, pages 192-199, New York, NY, USA, 1998. Association for Computing Machinery. URL: https://doi.org/10.1145/276884.276906.
  12. Artur Andrzejak and Emo Welzl. In between k-sets, j-facets, and i-faces: (i,j)-partitions. Discrete Comput. Geom., 29(1):105-131, 2003. Google Scholar
  13. Martin Balko, Radoslav Fulek, and Jan Kynčl. Crossing numbers and combinatorial characterization of monotone drawings of K_n. Discrete Comput. Geom., 53(1):107-143, 2015. Google Scholar
  14. József Balogh, Bernard Lidický, and Gelasio Salazar. Closing in on Hill’s conjecture. SIAM J. Discrete Math., 33(3):1261-1276, 2019. URL: https://doi.org/10.1137/17M1158859.
  15. József Balogh and Gelasio Salazar. k-sets, convex quadrilaterals, and the rectilinear crossing number of K_n. Discrete Comput. Geom., 35(4):671-690, 2006. URL: https://doi.org/10.1007/S00454-005-1227-6.
  16. Ranita Biswas, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. Depth in arrangements: Dehn-Sommerville-Euler relations with applications. J. Appl. Comput. Topol., 8(3):557-578, 2024. URL: https://doi.org/10.1007/S41468-024-00173-W.
  17. Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler. Oriented matroids, volume 46 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, 1999. Google Scholar
  18. Kenneth L. Clarkson and Peter W. Shor. Applications of random sampling in computational geometry. II. Discrete Comput. Geom., 4(5):387-421, 1989. URL: https://doi.org/10.1007/BF02187740.
  19. Tamal K. Dey. Improved bounds for planar k-sets and related problems. Discrete Comput. Geom., 19(3):373-382, 1998. URL: https://doi.org/10.1007/PL00009354.
  20. Jürgen Eckhoff. Helly, Radon, and Carathéodory type theorems. In Handbook of convex geometry, Vol. A, B, pages 389-448. North-Holland, Amsterdam, 1993. Google Scholar
  21. Herbert Edelsbrunner. Algorithms in combinatorial geometry, volume 10 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, 1987. URL: https://doi.org/10.1007/978-3-642-61568-9.
  22. Paul Erdős, László. Lovász, A. Simmons, and Ernst G. Straus. Dissection graphs of planar point sets. In A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), pages 139-149. North-Holland, Amsterdam-London, 1973. Google Scholar
  23. Branko Grünbaum. Convex polytopes, volume 221 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2003. Google Scholar
  24. T. Gulliksen and A. Hole. On the number of linearly separable subsets of finite sets in Rⁿ. Discrete Comput. Geom., 18(4):463-472, 1997. Google Scholar
  25. Carl W. Lee. Winding numbers and the generalized lower-bound conjecture. In Discrete and computational geometry (New Brunswick, NJ, 1989/1990), volume 6 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 209-219. Amer. Math. Soc., Providence, RI, 1991. URL: https://doi.org/10.1090/DIMACS/006/13.
  26. Johann Linhart. The upper bound conjecture for arrangements of halfspaces. Beiträge Algebra Geom., 35(1):29-35, 1994. Google Scholar
  27. Johann Linhart, Yanling Yang, and Martin Philipp. Arrangements of hemispheres and halfspaces. Discrete Math., 223(1-3):217-226, 2000. URL: https://doi.org/10.1016/S0012-365X(98)00360-4.
  28. László Lovász. On the number of halving lines. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 14:107-108, 1971. Google Scholar
  29. László Lovász, Katalin Vesztergombi, Uli Wagner, and Emo Welzl. Convex quadrilaterals and k-sets. In Towards a theory of geometric graphs, volume 342 of Contemp. Math., pages 139-148. Amer. Math. Soc., Providence, RI, 2004. Google Scholar
  30. Jiří Matoušek. Lectures on discrete geometry, volume 212 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2002. Google Scholar
  31. Peter McMullen. The maximum numbers of faces of a convex polytope. Mathematika, 17:179-184, 1970. Google Scholar
  32. John W. Moon. On the distribution of crossings in random complete graphs. J. Soc. Indust. Appl. Math., 13:506-510, 1965. Google Scholar
  33. Ketan Mulmuley. Dehn-sommerville relations, upper bound theorem, and levels in arrangements. In Chee Yap, editor, Proceedings of the Ninth Annual Symposium on Computational Geometry (San Diego, CA, USA, May 19-21, 1993), pages 240-246, 1993. URL: https://doi.org/10.1145/160985.161141.
  34. Petra Mutzel and Lutz Oettershagen. The crossing number of seq-shellable drawings of complete graphs. In Combinatorial algorithms, volume 10979 of Lecture Notes in Comput. Sci., pages 273-284. Springer, Cham, 2018. URL: https://doi.org/10.1007/978-3-319-94667-2_23.
  35. Arnau Padrol. Many neighborly polytopes and oriented matroids. Discrete Comput. Geom., 50(4):865-902, 2013. URL: https://doi.org/10.1007/S00454-013-9544-7.
  36. G. W. Peck. On "k-sets" in the plane. Discrete Math., 56(1):73-74, 1985. Google Scholar
  37. Marcus Schaefer. Crossing numbers of graphs. Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL, 2018. Google Scholar
  38. Edward R. Scheinerman and Herbert S. Wilf. The rectilinear crossing number of a complete graph and Sylvester’s "four point problem" of geometric probability. Amer. Math. Monthly, 101(10):939-943, 1994. Google Scholar
  39. Richard P. Stanley. The number of faces of a simplicial convex polytope. Adv. in Math., 35(3):236-238, 1980. Google Scholar
  40. Elizaveta Streltsova and Uli Wagner. Linear relations between face numbers of levels in arrangements. Preprint, 2025. URL: https://arxiv.org/abs/2504.07752.
  41. Elizaveta Streltsova and Uli Wagner. Sublevels in arrangements and the spherical arc crossing number of complete graphs. Preprint, 2025. URL: https://arxiv.org/abs/2504.07770.
  42. László A. Székely. Turán’s brick factory problem: the status of the conjectures of Zarankiewicz and Hill. In Graph theory - Favorite conjectures and open problems. 1, Probl. Books in Math., pages 211-230. Springer, 2016. Google Scholar
  43. Géza Tóth. Point sets with many k-sets. Discrete Comput. Geom., 26(2):187-194, 2001. URL: https://doi.org/10.1007/S004540010022.
  44. Uli Wagner. On a geometric generalization of the upper bound theorem. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pages 635-645, 2006. URL: https://doi.org/10.1109/FOCS.2006.53.
  45. Uli Wagner. k-Sets and k-facets. In Surveys on Discrete and Computational Geometry, volume 453 of Contemp. Math., pages 443-513. Amer. Math. Soc., Providence, RI, 2008. Google Scholar
  46. Emo Welzl. Entering and leaving j-facets. Discrete Comput. Geom., 25(3):351-364, 2001. URL: https://doi.org/10.1007/S004540010085.
  47. James G. Wendel. A problem in geometric probability. Math. Scand., 11:109-111, 1962. Google Scholar
  48. Günter M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact