  The main result of this paper is, that for convex billiards in higher
dimensions, in contrast with 2D case, 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 "transversal" variations.
