Maximizing orbits for higher dimensional convex billiards

TL;DR

This paper demonstrates that in higher-dimensional convex billiards, for any boundary point and any collision number, there are trajectories with conjugate points, highlighting a key difference from the 2D case.

Contribution

It establishes a variational property of higher-dimensional billiard orbits, showing that locally maximizing segments cannot approach the boundary too closely, unlike in 2D.

Findings

Existence of trajectories with conjugate points at any collision in higher dimensions.

Locally maximizing segments are bounded away from the boundary.

Different behavior of the second variation formula in longitudinal and transversal directions.

Abstract

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.