Estimation of the length of a simple geodesic on a convex surface

It was proved by I. M. Liberman that for a $C^2$-smooth closed surface $M$ of positive Gaussian curvature there exists a number $l$ such that any geodesic arc on $M$ of length at least $l$ is not simple. In this article we indicate a lower bound for $l$. We exhibit an example showing that our estimate is unimprovable.