Discrete & Computational Geometry

, Volume 47, Issue 2, pp 275–287 | Cite as

Illuminating Spindle Convex Bodies and Minimizing the Volume of Spherical Sets of Constant Width

  • Károly BezdekEmail author


A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a “fat” one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm’s theorem and its proof on illuminating convex bodies of constant width to the family of “fat” spindle convex bodies. Also, this leads to the spherical analog of the well-known Blaschke–Lebesgue problem.


Boltyanski–Hadwiger illumination conjecture (Fat) spindle convex bodies Spindle convex hull Gauss (resp., normal) images of faces Illumination by random directions Convex bodies of constant width in spherical space 


  1. 1.
    Aigner, M., Ziegler, G.M.: Proofs from The Book, 4th edn. Springer, Berlin, (2010) CrossRefGoogle Scholar
  2. 2.
    Allendoerfer, C.B.: Steiner’s formulae on a general S n+1. Bull. Am. Math. Soc. 54, 128–135 (1948) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bayen, T., Lachand-Robert, T., Oudet, É.: Analytic parametrization of three-dimensional bodies of constant width. Arch. Ration. Mech. Anal. 186(2), 225–249 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bezdek, K.: The illumination conjecture and its extensions. Period. Math. Hung. 53(1–2), 59–69 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bezdek, K., Kiss, Gy.: On the X-ray number of almost smooth convex bodies and of convex bodies of constant width. Can. Math. Bull. 52(3), 342–348 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bezdek, K., Lángi, Zs., Naszódi, M., Papez, P.: Ball-polyhedra. Discrete Comput. Geom. 38(2), 201–230 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Blaschke, W.: Einige Bemerkungen über Kurven und Flächen konstanter Breite. Leipz. Ber. 57, 290–297 (1915) Google Scholar
  8. 8.
    Boltyanski, V.: The problem of illuminating the boundary of a convex body. Izv. Mold. Fil. Akad. Nauk SSSR 76, 77–84 (1960) Google Scholar
  9. 9.
    Boltyanski, V.: Solution of the illumination problem for three dimensional convex bodies. Dokl. Akad. Nauk SSSR 375, 298–301 (2000) Google Scholar
  10. 10.
    Dekster, B.V.: Completeness and constant width in spherical and hyperbolic spaces. Acta Math. Hung. 67(4), 289–300 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dekster, B.V.: The Jung theorem for spherical and hyperbolic spaces. Acta Math. Hung. 67(4), 315–331 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Dumer, I.: Covering spheres with spheres. Discrete Comput. Geom. 38, 665–679 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Erdős, P., Rogers, C.A.: Covering space with convex bodies. Acta Arith. 7, 281–285 (1962) MathSciNetGoogle Scholar
  14. 14.
    Erdős, P., Rogers, C.A.: The star number of coverings of space with convex bodies. Acta Arith. 9, 41–45 (1964) MathSciNetGoogle Scholar
  15. 15.
    Fowler, P.W., Tarnai, T.: Transition from spherical circle packing to covering: geometrical analogues of chemical isomerization. Proc. R. Soc. Lond. 452, 2043–2064 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Hadwiger, H.: Ungelöste Probleme, Nr. 38. Elem. Math. 15, 130–131 (1960) MathSciNetGoogle Scholar
  17. 17.
    Kahn, J., Kalai, G.: A counterexample to Borsuk’s conjecture. Bull., New Ser., Am. Math. Soc. 29(1), 60–62 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Lassak, M.: Covering the boundary of a convex set by tiles. Proc. Am. Math. Soc. 104, 269–272 (1988) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Martini, H., Soltan, V.: Combinatorial problems on the illumination of convex bodies. Aequ. Math. 57, 121–152 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Meissner, E.: Über Punktmengen konstanter Breite. Vjschr. Naturforsch. Ges. Zurich 56, 42–50 (1911) Google Scholar
  21. 21.
    Papadoperakis, I.: An estimate for the problem of illumination of the boundary of a convex body in E 3. Geom. Dedic. 75, 275–285 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Rogers, C.A., Shephard, G.C.: The difference body of a convex body. Arch. Math. 8, 220–233 (1957) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Santaló, L.A.: On parallel hypersurfaces in the elliptic and hyperbolic n-dimensional space. Proc. Am. Math. Soc. 1, 325–330 (1950) CrossRefzbMATHGoogle Scholar
  24. 24.
    Santaló, L.A.: Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and its Applications, vol. 1. Addison-Wesley, Reading (1976) zbMATHGoogle Scholar
  25. 25.
    Schramm, O.: Illuminating sets of constant width. Mathematika 35, 180–189 (1988) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada
  2. 2.Department of MathematicsUniversity of PannoniaVeszprémHungary
  3. 3.Institute of MathematicsEötvös UniversityBudapestHungary

Personalised recommendations