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Geometric and Functional Analysis

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

Periodic Billiard Trajectories in Smooth Convex Bodies

  • Roman N. Karasev
Article

Abstract

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 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

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

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