Skip to main content
SpringerLink
Log in

On the Volume of Boolean Expressions of Balls – A Review of the Kneser–Poulsen Conjecture

  • Chapter
  • First Online:
New Trends in Intuitive Geometry

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 27))

Abstract

In 1954–55, E. T. Poulsen and M. Kneser formulated the conjecture that if some congruent balls of the Euclidean space are rearranged in such a way that the distances between the centers of the balls do not increase, then the volume of the union of the balls does not increase as well. Our goal is to give a survey of attempts to prove this conjecture, to discuss possible generalizations, and to collect some relevant open questions.

The author was supported by the Hungarian Scientific Research Fund (OTKA), Grant No. K112703.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. E.T. Poulsen, Problem 10. Math. Scand. 2, 346 (1954)

    Google Scholar 

  2. M. Kneser, Einige Bemerkungen über das Minkowskische Flächenmass. Arch. Math. (Basel) 6, 382–390 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Federer, in Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153 (Springer, New York Inc., 1969)

    Google Scholar 

  4. A. Kolmogoroff, Beiträge zur Maßtheorie. Math. Ann. 107, 351–366 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Hadwiger, Ungelöste probleme Nr. 11. Elem. Math. 11, 51–60 (1956)

    MathSciNet  Google Scholar 

  6. V. Klee, Some unsolved problems in plane geometry. Math. Mag. 52(3), 131–145 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. W. Moser, Problem 32. Pushing disks around, in Research Problems in Discrete Geometry, 6th edn. (1981), pp. 1–32[mimeographed notes]

    Google Scholar 

  8. V. Klee, S. Wagon, in Old and new unsolved problems in plane geometry and number theory, vol. 11, The Dolciani mathematical expositions (Mathematical Association of America, Washington, 1991)

    Google Scholar 

  9. K. Bezdek, in Classical Topics in Discrete Geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC (Springer, New York, 2010)

    Book  MATH  Google Scholar 

  10. K. Bezdek, Ensembles impropres et nombre dimensionnel I. Bull. Sci. Math., II. Sér. 52, 320–344 (1928)

    MATH  Google Scholar 

  11. D. Avis, B.K. Bhattacharya, H. Imai, Computing the volume of the union of spheres. Vis. Comput. 3, 323–328 (1988)

    Article  MATH  Google Scholar 

  12. W. Rehder, On the volume of unions of translates of a convex set. Am. Math. Mon. 87(5), 382–384 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Bollobás, Area of the union of disks. Elem. Math. 23, 60–61 (1968)

    MathSciNet  MATH  Google Scholar 

  14. B. Csikós, “Körrendszer által lefedett tartomány területének megváltozása a körök mozgatása esetén,” in A XVI. Országos Tudományos Diákköri Konferencia kiemelkedő pályamunkái III., vol. 6 of Bolyai Soc. Math. Stud., pp. 209–212, Művelődési Minisztérium Tudományszervezési és Informatikai Intézete (1984)

    Google Scholar 

  15. B. Csikós, On the Hadwiger–Kneser–Poulsen conjecture, in Intuitive Geometry (Budapest, 1995). Bolyai Society Mathematical Studies, vol. 6 (János Bolyai Mathematical Society, Budapest, 1997), pp. 291–299

    Google Scholar 

  16. M. Bern, A. Sahai, Pushing disks together–the continuous-motion case. Discret. Comput. Geom. 20(4), 499–514 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. B. Csikós, On the volume of the union of balls. Discret. Comput. Geom. 20(4), 449–461 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Gromov, Monotonicity of the volume of intersection of balls, in Geometrical Aspects of Functional Analysis (1985–86). Lecture Notes in Mathematics, vol. 1267 (Springer, Berlin, 1987), pp.1–4

    Google Scholar 

  19. Y. Gordon, M. Meyer, On the volume of unions and intersections of balls in Euclidean space, in Geometric Aspects of Functional Analysis (Israel, 1992–1994), Operator theory advance application, vol. 77 (Birkhäuser, Basel, 1995), pp. 91–101

    Chapter  MATH  Google Scholar 

  20. B. Csikós, On the volume of flowers in space forms. Geom. Dedicata 86(1–3), 59–79 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. H. Cheng, S.P. Tan, Y. Zheng, On continuous expansions of configurations of points in Euclidean space, arXiv:1107.0140v1 [math.MG] (2011), pp. 1–9

  22. T. Kato, Perturbation Theory for Linear Operators. Classics in Mathematics (Springer, Berlin, 1995) [Reprint of the 1980 edition]

    Book  MATH  Google Scholar 

  23. R. Alexander, Lipschitzian mappings and total mean curvature of polyhedral surfaces I. Trans. Am. Math. Soc. 288(2), 661–678 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. V. Capoyleas, J. Pach, On the perimeter of a point set in the plane, in Discrete and Computational Geometry (New Brunswick, NJ, 1989–1990), DIMACS series discrete mathematics and theoretical computer science, vol. 6 (American Mathematical Society, Providence, RI, 1991), pp. 67–76

    Google Scholar 

  25. I. Gorbovickis, Strict Kneser–Poulsen conjecture for large radii. Geom. Dedicata 162, 95–107 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. D. Gale, On Lipschitzian mappings of convex bodies, in Proceedings of Symposia in Pure Mathematics, vol. VII (American Mathematical Society, Providence, 1963), , pp. 221–223

    Google Scholar 

  27. K. Bezdek, R. Connelly, The Kneser–Poulsen conjecture for spherical polytopes. Discret. Comput. Geom. 32(1), 101–106 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. K. Bezdek, On the monotonicity of the volume of hyperbolic convex polyhedra. Beiträge Algebr. Geom. 46(2), 609–614 (2005)

    MathSciNet  MATH  Google Scholar 

  29. K. Bezdek, R. Connelly, Pushing disks apart–the Kneser–Poulsen conjecture in the plane. J. Reine Angew. Math. 553, 221–236 (2002)

    MathSciNet  MATH  Google Scholar 

  30. I. Gorbovickis, Kneser–Poulsen conjecture for a small number of intersections. Contrib. Discret. Math. 9(1), 1–10 (2014)

    MathSciNet  MATH  Google Scholar 

  31. K. Bezdek, R. Connelly, On the weighted Kneser–Poulsen conjecture. Period. Math. Hungar. 57(2), 121–129 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. I. Rivin, J.-M. Schlenker, The Schläfli formula in Einstein manifolds with boundary. Electron. Res. Announc. Am. Math. Soc. 5, 18–23 (1999). electronic

    Article  MATH  Google Scholar 

  33. R. Souam, The Schläfli formula for polyhedra and piecewise smooth hypersurfaces. Differ. Geom. Appl. 20(1), 31–45 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. B. Csikós, A Schläfli-type formula for polytopes with curved faces and its application to the Kneser–Poulsen conjecture. Monatsh. Math. 147(4), 273–292 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. A.L. Besse, in Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10 [Results in Mathematics and Related Areas (3)] (Springer, Berlin, 1987)

    Google Scholar 

  36. S. Lang, in Fundamentals of Differential Geometry. Graduate Texts in Mathematics , vol. 191 (Springer, New York, 1999)

    Book  MATH  Google Scholar 

  37. K. Bezdek, R. Connelly, B. Csikós, On the perimeter of the intersection of congruent disks. Beiträge Algebr. Geom. 47(1), 53–62 (2006)

    MathSciNet  MATH  Google Scholar 

  38. M. Meyer, S. Reisner, M. Schmuckenschläger, The volume of the intersection of a convex body with its translates. Mathematika 40(2), 278–289 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  39. I. Gorbovickis, The central set and its application to the Kneser–Poulsen conjecture. Discrete Comput. Geom. 59(4), 784–801 (June 2018), arXiv:1511.08134v2 [math.MG]

    Article  MathSciNet  MATH  Google Scholar 

  40. O. Kowalski, L. Vanhecke, Ball-homogeneous and disk-homogeneous Riemannian manifolds. Math. Z. 180(4), 429–444 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  41. G. Calvaruso, L. Vanhecke, Special ball-homogeneous spaces. Z. Anal. Anwend. 16(4), 789–800 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  42. G. Calvaruso, P. Tondeur, L. Vanhecke, Four-dimensional ball-homogeneous and \(C\)-spaces. Beiträge Algebr. Geom. 38(2), 325–336 (1997)

    MathSciNet  MATH  Google Scholar 

  43. A. Gray, L. Vanhecke, Riemannian geometry as determined by the volumes of small geodesic balls. Acta Math. 142(3–4), 157–198 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  44. B. Csikós, M. Horváth, On the volume of the intersection of two geodesic balls. Differ. Geom. Appl. 29(4), 567–576 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. B. Csikós, M. Horváth, A characterization of harmonic spaces. J. Differ. Geom. 90(3), 383–389 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  46. E. Copson, H. Ruse, Harmonic Riemannian spaces. Proc. R. Soc. Edinb. 60, 117–133 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  47. B. Csikós, D. Kunszenti-Kovács, On the extendability of the Kneser–Poulsen conjecture to Riemannian manifolds. Adv. Geom. 10(2), 197–204 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. B. Csikós, M. Horváth, A characterization of spaces of constant curvature by minimum covering radius of triangles. Indag. Math. (N.S.) 25(3), 608–617 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  49. B. Csikós, G. Moussong, On the Kneser–Poulsen conjecture in elliptic space. Manuscr. Math. 121(4), 481–489 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Eötvös University, Pázmány Péter sétány 1/C, Budapest, 1117, Hungary

    Balázs Csikós

Authors
  1. Balázs Csikós
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Balázs Csikós .

Editor information

Editors and Affiliations

  1. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gergely Ambrus

  2. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Imre Bárány

  3. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Károly J. Böröczky

  4. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gábor Fejes Tóth

  5. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    János Pach

Rights and permissions

Reprints and permissions

Copyright information

© 2018 János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Csikós, B. (2018). On the Volume of Boolean Expressions of Balls – A Review of the Kneser–Poulsen Conjecture. In: Ambrus, G., Bárány, I., Böröczky, K., Fejes Tóth, G., Pach, J. (eds) New Trends in Intuitive Geometry. Bolyai Society Mathematical Studies, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57413-3_4

Download citation

Publish with us

Policies and ethics

Search

Navigation