Periodic points of the billiard ball map in a convex domain

We study periodic orbits of the billiard ball map in a strictly convex domain with a C^∞ boundary. We conjecture that the Lebesgue measure of all periodic points is 0. We are able to prove the following partial result: the measure of period three periodic orbits is 0. The question of whether the mentioned measure is 0 appeared in the study of spectral invariants of a planar region. The author learned about it from R. Melrose.