Skip to main content
SpringerLink
Log in

Quantitative Combinatorial Geometry for Continuous Parameters

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

We prove variations of Carathéodory’s, Helly’s and Tverberg’s theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lovász’s colorful Helly’s theorem, Bárány’s colorful Carathéodory’s theorem, and the colorful Tverberg’s theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aliev, I., Bassett, R., De Loera, J.A., Louveaux, Q.: A quantitative Doignon–Bell–Scarf theorem. Combinatorica (2016). doi:10.1007/s00493-015-3266-9

  2. Alon, N., Kleitman, D.J.: Piercing convex sets and the Hadwiger–Debrunner (\(p, q\))-problem. Adv. Math. 96(1), 103–112 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amenta, N., De Loera, J.A., Soberón, P.: Helly’s theorem: new variations and applications. arXiv:1508.07606 (2015)

  4. Arocha, J.L., Bárány, I., Bracho, J., Fabila, R., Montejano, L.: Very colorful theorems. Discrete Comput. Geom. 42(2), 142–154 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bárány, I.: A generalization of Carathéodory’s theorem. Discrete Math. 40(2–3), 141–152 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bárány, I., Larman, D.G.: A colored version of Tverberg’s theorem. J. Lond. Math. Soc. 2(2), 314–320 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bárány, I., Onn, S.: Colourful linear programming and its relatives. Math. Oper. Res. 22(3), 550–567 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bárány, I., Katchalski, M., Pach, J.: Quantitative Helly-type theorems. Proc. Am. Math. Soc. 86(1), 109–114 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bárány, I., Katchalski, M., Pach, J.: Helly’s theorem with volumes. Am. Math. Monthly 91(6), 362–365 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. Barvinok, A.: Thrifty approximations of convex bodies by polytopes. Int. Math. Res. Not. 2014(16), 4341–4356 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Blagojević, P.V.M., Matschke, B., Ziegler, G.M.: Optimal bounds for a colorful Tverberg–Vrećica type problem. Adv. Math. 226(6), 5198–5215 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Blagojević, P.V.M., Matschke, B., Ziegler, G.M.: Optimal bounds for the colored Tverberg problem. J. Eur. Math. Soc. 17, 739–754 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Blaschke, W.: Affine Differentialgeometrie. Springer, Berlin (1923)

    MATH  Google Scholar 

  14. Böröczky Jr., K.: Approximation of general smooth convex bodies. Adv. Math. 153(2), 325–341 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bronstein, E.M.: Approximation of convex sets by polytopes. J. Math. Sci. 153(6), 727–762 (2008)

    Article  MATH  Google Scholar 

  16. Bronstein, E.M., Ivanov, L.D.: The approximation of convex sets by polyhedra (Russian). Sibirsk. Mat. Ž. 16(5), 1110–1112, 1132 (1975)

  17. Carathéodory, C.: Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann. 64(1), 95–115 (1907)

    Article  MathSciNet  MATH  Google Scholar 

  18. Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Proceedings of Symposia in Pure Mathematics, vol. VII, pp. 101–180. American Mathematical Society, Providence, RI (1963)

  19. De Loera, J.A., La Haye, R.N., Rolnick, D., Soberón, P.: Quantitative Tverberg, Helly & Carathéodory theorems. arXiv:1503.06116 (2015)

  20. De Loera, J.A., La Haye, R.N., Rolnick, D., Soberón, P.: Quantitative Tverberg theorems over lattices and other discrete sets (2016)

  21. Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory 10(3), 227–236 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  22. Eckhoff, J.: Helly, Radon, and Carathéodory type theorems. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. A, B, pp. 389–448. North-Holland, Amsterdam (1993)

    Chapter  Google Scholar 

  23. Frick, F.: Counterexamples to the topological Tverberg conjecture. arXiv:1502.00947 (2015)

  24. Gordon, Y., Meyer, M., Reisner, S.: Constructing a polytope to approximate a convex body. Geom. Dedicata 57(2), 217–222 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gordon, Y., Reisner, S., Schütt, C.: Umbrellas and polytopal approximation of the euclidean ball. J. Approx. Theory 90(1), 9–22 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gruber, P.M.: Aspects of approximation of convex bodies. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. A, B, pp. 319–345. North-Holland, Amsterdam (1993)

    Chapter  Google Scholar 

  27. Gruber, P.M.: Asymptotic estimates for best and stepwise approximation of convex bodies. I. Forum Math. 5(3), 281–297 (1993)

    MathSciNet  MATH  Google Scholar 

  28. Helly, E.: Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahresber. Dtsch. Math. Ver. 32, 175–176 (1923)

    MATH  Google Scholar 

  29. Helly, E.: Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten. Monatsh. Math. Phys. 37(1), 281–302 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  30. Holmsen, A.F., Pach, J., Tverberg, H.: Points surrounding the origin. Combinatorica 28(6), 633–644 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Katchalski, M., Liu, A.: A problem of geometry in \({\mathbb{R}}^n\). Proc. Am. Math. Soc. 75(2), 284–288 (1979)

    MATH  Google Scholar 

  32. Kirkpatrick, D., Mishra, B., Yap, C.-K.: Quantitative Steinitz’s theorems with applications to multifingered grasping. Discrete Comput. Geom. 7(3), 295–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  33. Langberg, M., Schulman, L.J.: Contraction and expansion of convex sets. Discrete Comput. Geom. 42(4), 594–614 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mabillard, I., Wagner, U.: Eliminating Tverberg points. I. An analogue of the Whitney trick. In: Proceedings of the 30th Annual Symposium on Computational Geometry (SoCG), Kyoto, June 2014, pp. 171–180. ACM, New York (2014)

  35. Matoušek, J.: Lectures on discrete geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)

    Book  MATH  Google Scholar 

  36. Naszódi, M.: Proof of a conjecture of Bárány, Katchalski and Pach. Discrete Comput. Geom. 55(1), 243–248 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Radon, J.: Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Math. Ann. 83(1–2), 113–115 (1921)

    Article  MathSciNet  MATH  Google Scholar 

  38. Reisner, S., Schütt, C., Werner, E.: Dropping a vertex or a facet from a convex polytope. Forum Math. 13, 359–378 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rolnick, D., Soberón, P.: Quantitative \((p,q)\) theorems in combinatorial geometry. arXiv:1504.01642 (2015)

  40. Roudneff, J.-P.: Partitions of points into simplices with \(k\)-dimensional intersection. I. The conic Tverberg’s theorem. Eur. J. Combin. 22(5), 733–743 (2001). Combinatorial geometries (Luminy, 1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sarkaria, K.S.: Tverberg’s theorem via number fields. Isr. J. Math. 79(2), 317–320 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  42. Soberón P.: Equal coefficients and tolerance in coloured Tverberg partitions. In: Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry, pp. 91–96. ACM, New York (2013)

  43. Steinitz E.: Bedingt konvergente Reihen und konvexe Systeme. J. Reine Angew. Math., 143, 128–175 (1913); 144, 1–40 (1914); 146, 1-52 (1916)

  44. Tverberg, H.: A generalization of Radon’s theorem. J. Lond. Math. Soc. 41(1), 123–128 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tverberg, H.: A generalization of Radon’s theorem. II. Bull. Aust. Math. Soc. 24(3), 321–325 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wenger, R.: Helly-type theorems and geometric transversals. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 63–82. CRC Press, Boca Raton (1997)

    Google Scholar 

  47. Ziegler, G.M.: 3N colored points in a plane. Not. Am. Math. Soc. 58(4), 550–557 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to I. Bárány, A. Barvinok, F. Frick, A. Holmsen, J. Pach, and G.M. Ziegler for their comments and suggestions. This work was partially supported by the Institute for Mathematics and its Applications (IMA) in Minneapolis, MN funded by the National Science Foundation (NSF). The authors are grateful for the wonderful working environment that led to this paper. The research of De Loera and La Haye was also supported by a UC MEXUS grant. De Loera was also supported by NSF Grant DMS-1522158. Rolnick was additionally supported by NSF Grants DMS-1321794 and 1122374.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, Davis, CA, 95616, USA

    Jesús A. De Loera & Reuben N. La Haye

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    David Rolnick

  3. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Pablo Soberón

Authors
  1. Jesús A. De Loera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reuben N. La Haye
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Rolnick
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pablo Soberón
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pablo Soberón.

Additional information

Editor in Charge: Günter M. Ziegler

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

De Loera, J.A., La Haye, R.N., Rolnick, D. et al. Quantitative Combinatorial Geometry for Continuous Parameters. Discrete Comput Geom 57, 318–334 (2017). https://doi.org/10.1007/s00454-016-9857-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-016-9857-4

Keywords

Search

Navigation