Maximizing orbits for higher-dimensional convex billiards

The main result of this paper is that, in contrast to the 2D case, for convex billiards in higher dimensions, for every point on the boundary, and for every $n$, there always exist billiard trajectories developing conjugate points at the $n$-th collision with the boundary. We shall explain that this is a consequence of the following variational property of the billiard orbits in higher dimension. If a segment of an orbit is locally maximizing, then it can not pass too close to the boundary. This fact follows from the second variation formula for the length functional. It turns out that this formula behaves differently with respect to "longitudinal'' and "transverse'' variations.