View-obstruction and a conjecture of Schoenberg

Here a conjecture of Schoenberg regarding the billiard ball problem for spheres is proved in the Euclidean space R(n) for n = 3, 4. Markoff type chains of related isolated extreme values are also obtained. This is achieved by using the theory of results about covering radii of lattices in the plane and in R(3). Analogous results for (n - 2)-dimensional trajectories in R(n), for all n greater than or equal to 3, are also obtained.