  We provide lower bounds on the number of periodic Finsler billiard
trajectories inside a quadratically convex smooth closed hypersurface $M$ in a
$d$-dimensional Finsler space with possibly irreversible Finsler metric. An
example of such a system is a billiard in a sufficiently weak magnetic field.
The $r$-periodic Finsler billiard trajectories correspond to $r$-gons inscribed
in $M$ and having extremal Finsler length. The cyclic group ${\mathbb Z}_r$
acts on these extremal polygons, and one counts the ${\mathbb Z}_r$-orbits.
Using Morse and Lusternik-Schnirelmann theories, we prove that if $r\ge 3$ is
prime, then the number of $r$-periodic Finsler billiard trajectories is not
less than $(r-1)(d-2)+1$. We also give stronger lower bounds when $M$ is in
general position. The problem of estimating the number of periodic billiard
trajectories from below goes back to Birkhoff. Our work extends to the Finsler
setting the results previously obtained for Euclidean billiards by Babenko,
Farber, Tabachnikov, and Karasev.
