Topology of billiard problems, I

Let T ⊂ R^m+1 be a strictly convex domain bounded by a smooth hypersurface X = ∂T. In this paper we find lower bounds on the number of billiard trajectories in T which have a prescribed initial point A ∈ X, a prescribed final point B ∈ X, and make a prescribed number n of reflections at the boundary X. We apply a topological approach based on the calculation of cohomology rings of certain configuration spaces of S^m.