  In this paper we show that in some cases the E.Hopf rigidity phenomenon
admits quantitative interpretation. More precisely we estimate from above the
measure of the set $\mathcal{M}$ swept by minimal orbits. These estimates are
sharp, i.e. if $\mathcal{M}$ occupies the whole phase space we recover the
E.Hopf rigidity. We give these estimates in two cases: the first is the case of
convex billiards in the plane, sphere or hyperbolic plane. The second is the
case of conformally flat Riemannian metrics on a torus. It seems to be a
challenging question to understand such a quantitative bounds for Burago-Ivanov
theorem.
