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Abstract
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We consider two nested billiards in
,
, 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 which says that in
higher dimensions only quadrics may have caustics. We also prove versions of
Berger’s theorem and the main result for billiards in spaces of constant curvature
(space forms).
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Keywords
commuting billiards, caustics, space forms, confocal
quadrics
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Mathematical Subject Classification 2010
Primary: 70H99
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Milestones
Received: 24 August 2018
Revised: 23 August 2019
Accepted: 3 September 2019
Published: 29 April 2020
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