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

  • Balázs CsikósEmail author
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 27)


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.


Kneser–Poulsen conjecture Volume inequalities Variation of the volume Schläfli formula 

2010 Mathematics Subject Classification

52A40 52A38 26B20 


  1. 1.
    E.T. Poulsen, Problem 10. Math. Scand. 2, 346 (1954)Google Scholar
  2. 2.
    M. Kneser, Einige Bemerkungen über das Minkowskische Flächenmass. Arch. Math. (Basel) 6, 382–390 (1955)MathSciNetzbMATHGoogle Scholar
  3. 3.
    H. Federer, in Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153 (Springer, New York Inc., 1969)Google Scholar
  4. 4.
    A. Kolmogoroff, Beiträge zur Maßtheorie. Math. Ann. 107, 351–366 (1932)MathSciNetzbMATHGoogle Scholar
  5. 5.
    H. Hadwiger, Ungelöste probleme Nr. 11. Elem. Math. 11, 51–60 (1956)MathSciNetGoogle Scholar
  6. 6.
    V. Klee, Some unsolved problems in plane geometry. Math. Mag. 52(3), 131–145 (1979)MathSciNetzbMATHGoogle Scholar
  7. 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. 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. 9.
    K. Bezdek, in Classical Topics in Discrete Geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC (Springer, New York, 2010)zbMATHGoogle Scholar
  10. 10.
    K. Bezdek, Ensembles impropres et nombre dimensionnel I. Bull. Sci. Math., II. Sér. 52, 320–344 (1928)zbMATHGoogle Scholar
  11. 11.
    D. Avis, B.K. Bhattacharya, H. Imai, Computing the volume of the union of spheres. Vis. Comput. 3, 323–328 (1988)zbMATHGoogle Scholar
  12. 12.
    W. Rehder, On the volume of unions of translates of a convex set. Am. Math. Mon. 87(5), 382–384 (1980)MathSciNetzbMATHGoogle Scholar
  13. 13.
    B. Bollobás, Area of the union of disks. Elem. Math. 23, 60–61 (1968)MathSciNetzbMATHGoogle Scholar
  14. 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. 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. 291–299Google Scholar
  16. 16.
    M. Bern, A. Sahai, Pushing disks together–the continuous-motion case. Discret. Comput. Geom. 20(4), 499–514 (1998)MathSciNetzbMATHGoogle Scholar
  17. 17.
    B. Csikós, On the volume of the union of balls. Discret. Comput. Geom. 20(4), 449–461 (1998)MathSciNetzbMATHGoogle Scholar
  18. 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.1–4Google Scholar
  19. 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. 91–101zbMATHGoogle Scholar
  20. 20.
    B. Csikós, On the volume of flowers in space forms. Geom. Dedicata 86(1–3), 59–79 (2001)MathSciNetzbMATHGoogle Scholar
  21. 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. 22.
    T. Kato, Perturbation Theory for Linear Operators. Classics in Mathematics (Springer, Berlin, 1995) [Reprint of the 1980 edition]zbMATHGoogle Scholar
  23. 23.
    R. Alexander, Lipschitzian mappings and total mean curvature of polyhedral surfaces I. Trans. Am. Math. Soc. 288(2), 661–678 (1985)MathSciNetzbMATHGoogle Scholar
  24. 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. 67–76Google Scholar
  25. 25.
    I. Gorbovickis, Strict Kneser–Poulsen conjecture for large radii. Geom. Dedicata 162, 95–107 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    D. Gale, On Lipschitzian mappings of convex bodies, in Proceedings of Symposia in Pure Mathematics, vol. VII (American Mathematical Society, Providence, 1963), , pp. 221–223Google Scholar
  27. 27.
    K. Bezdek, R. Connelly, The Kneser–Poulsen conjecture for spherical polytopes. Discret. Comput. Geom. 32(1), 101–106 (2004)MathSciNetzbMATHGoogle Scholar
  28. 28.
    K. Bezdek, On the monotonicity of the volume of hyperbolic convex polyhedra. Beiträge Algebr. Geom. 46(2), 609–614 (2005)MathSciNetzbMATHGoogle Scholar
  29. 29.
    K. Bezdek, R. Connelly, Pushing disks apart–the Kneser–Poulsen conjecture in the plane. J. Reine Angew. Math. 553, 221–236 (2002)MathSciNetzbMATHGoogle Scholar
  30. 30.
    I. Gorbovickis, Kneser–Poulsen conjecture for a small number of intersections. Contrib. Discret. Math. 9(1), 1–10 (2014)MathSciNetzbMATHGoogle Scholar
  31. 31.
    K. Bezdek, R. Connelly, On the weighted Kneser–Poulsen conjecture. Period. Math. Hungar. 57(2), 121–129 (2008)MathSciNetzbMATHGoogle Scholar
  32. 32.
    I. Rivin, J.-M. Schlenker, The Schläfli formula in Einstein manifolds with boundary. Electron. Res. Announc. Am. Math. Soc. 5, 18–23 (1999). electroniczbMATHGoogle Scholar
  33. 33.
    R. Souam, The Schläfli formula for polyhedra and piecewise smooth hypersurfaces. Differ. Geom. Appl. 20(1), 31–45 (2004)MathSciNetzbMATHGoogle Scholar
  34. 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)MathSciNetzbMATHGoogle Scholar
  35. 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. 36.
    S. Lang, in Fundamentals of Differential Geometry. Graduate Texts in Mathematics , vol. 191 (Springer, New York, 1999)zbMATHGoogle Scholar
  37. 37.
    K. Bezdek, R. Connelly, B. Csikós, On the perimeter of the intersection of congruent disks. Beiträge Algebr. Geom. 47(1), 53–62 (2006)MathSciNetzbMATHGoogle Scholar
  38. 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)MathSciNetzbMATHGoogle Scholar
  39. 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]MathSciNetzbMATHGoogle Scholar
  40. 40.
    O. Kowalski, L. Vanhecke, Ball-homogeneous and disk-homogeneous Riemannian manifolds. Math. Z. 180(4), 429–444 (1982)MathSciNetzbMATHGoogle Scholar
  41. 41.
    G. Calvaruso, L. Vanhecke, Special ball-homogeneous spaces. Z. Anal. Anwend. 16(4), 789–800 (1997)MathSciNetzbMATHGoogle Scholar
  42. 42.
    G. Calvaruso, P. Tondeur, L. Vanhecke, Four-dimensional ball-homogeneous and \(C\)-spaces. Beiträge Algebr. Geom. 38(2), 325–336 (1997)MathSciNetzbMATHGoogle Scholar
  43. 43.
    A. Gray, L. Vanhecke, Riemannian geometry as determined by the volumes of small geodesic balls. Acta Math. 142(3–4), 157–198 (1979)MathSciNetzbMATHGoogle Scholar
  44. 44.
    B. Csikós, M. Horváth, On the volume of the intersection of two geodesic balls. Differ. Geom. Appl. 29(4), 567–576 (2011)MathSciNetzbMATHGoogle Scholar
  45. 45.
    B. Csikós, M. Horváth, A characterization of harmonic spaces. J. Differ. Geom. 90(3), 383–389 (2012)MathSciNetzbMATHGoogle Scholar
  46. 46.
    E. Copson, H. Ruse, Harmonic Riemannian spaces. Proc. R. Soc. Edinb. 60, 117–133 (1940)MathSciNetzbMATHGoogle Scholar
  47. 47.
    B. Csikós, D. Kunszenti-Kovács, On the extendability of the Kneser–Poulsen conjecture to Riemannian manifolds. Adv. Geom. 10(2), 197–204 (2010)MathSciNetzbMATHGoogle Scholar
  48. 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)MathSciNetzbMATHGoogle Scholar
  49. 49.
    B. Csikós, G. Moussong, On the Kneser–Poulsen conjecture in elliptic space. Manuscr. Math. 121(4), 481–489 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute of Mathematics, Eötvös UniversityBudapestHungary

Personalised recommendations