Periodic billiard trajectories in smooth convex bodies

TL;DR

This paper establishes a new lower bound on the number of distinct periodic billiard trajectories with a fixed number of reflections in smooth convex bodies, improving previous estimates especially for prime numbers of reflections.

Contribution

It provides a significantly improved lower bound for the count of periodic billiard trajectories with exactly p reflections in smooth convex bodies, especially for prime p.

Findings

Lower bound of (d-2)(p-1)+2 for prime p

Improved estimates over previous results

Applicable in $ ext{R}^d$ for smooth convex bodies

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.