On commuting billiards in higher dimensions

Alexey Glutsyuk

Published 2018-07-27Version 1

We consider two nested billiards in $\mathbb R^n$, $n\geq3$, with smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting Billiard Conjecture due to Sergei Tabachnikov. The main result is deduced from the classical theorem due to Marcel Berger saying that in higher dimensions only quadrics may have caustics.