Geometric and Functional Analysis

, Volume 19, Issue 2, pp 423–428 | Cite as

Periodic Billiard Trajectories in Smooth Convex Bodies

  • Roman N. KarasevEmail author


We consider billiard trajectories in a smooth convex body in \({\mathbb{R}^{d}}\) and estimate the number of distinct periodic trajectories that make exactly p reflections per period at the boundary of the body. In the case of prime p we obtain the lower bound (d – 2)(p – 1) + 2, which is much better than the previous estimates.

Keywords and phrases

Billiard trajectories Lyusternik–Schnirelmann theory 

2000 Mathematics Subject Classification

55M30 55M35 55R80 57S17 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. B.
    Babenko I.K.: Periodic trajectories in three-dimensional Birkhoff billiards. Math. Sbornik 181(9), 1155–1169 (1990)zbMATHGoogle Scholar
  2. Bi.
    G. Birkhoff, Dynamical Systems, Amer. Math. Soc. Coll. Publ. 9, 1927.Google Scholar
  3. FH.
    Fadell E., Husseini S.: Category weight and Steenrod operations. Boletin de la Sociedad Matemática Mexicana 37, 151–161 (1992)MathSciNetzbMATHGoogle Scholar
  4. Fa.
    Farber M.: Topology of billiard problems, II. Duke Mathematica Journal 115, 587–621 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. FaT.
    Farber M., Tabachnikov S.: Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards. Topology 41(3), 553–589 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  6. H.
    Hsiang W.Y.: Cohomology Theory of Topological Transformation Groups. Springer Verlag, Berlin-Heidelberg-New-York (1975)zbMATHGoogle Scholar
  7. KW.
    V. Klee, S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, Dolciani Mathematical Expositions, The Mathematical Association of America, 1996.Google Scholar
  8. Ku.
    Kuiper N.H.: Double normals of convex bodies. Israel Journal of Mathematics 2, 71–80 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  9. M.
    Makeev V.V.: Knaster’s problem and almost spherical sections. Math. Sbornik 180(3), 424–431 (1989)MathSciNetzbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Dept. of MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia

Personalised recommendations