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Santalo's formula and stability of trapping sets of positive measure

Billiard trajectories (broken generalised geodesics) are considered in the exterior of an obstacle $K$ with smooth boundary on an arbitrary Riemannian manifold. We prove a generalisation of the well-known Santalo's formula. As a consequence, it is established that if the set of trapped points has positive measure, then for all sufficiently small smooth perturbations of the boundary of $K$ the set of trapped points for the new obstacle obtained in this way also has positive measure. More generally the measure of the set of trapped points depends continuously on perturbations of the obstacle $K$. Some consequences of the generalised Santalo's formula are derived in the case of scattering by an obstacle $K$ in an Euclidean space. For example, it is shown that, for a large class of obstacles $K$, the volume of $K$ is uniquely determined by the average travelling times of scattering rays in the exterior of $K$.