Gruber P. Convex and Discrete Geometry

10 Problems of Minkowski and Weyl and Some Dynamics 201
C
Fig. 10.2. Gardener construction of a billiard table, given a caustic
Billiards have been considered from various viewpoints in the context of ergodic
theory, dynamical systems, partial differential equations, mechanics, physics, number
theory and geometry. For some information, see the books of Birkhoff [120],
Arnold [38], Lazutkin [632], Cornfeld, Fomin and Sinai [224], Katok, Strelcyn,
Ledrappier and Przytycki [567], Gal’perin and Zemlyakov [354], Kozlov and
Treshchëv [614], Petkov and Stoyanov [795], and Tabachnikov [985].
We consider convex caustics from a geometric viewpoint. These are convex
bodies C in int B such that any trajectory which touches C once, touches it again
after each reflection. Minasian [728] showed the following. Let C be a caustic of
a planar billiard table B. Then there is a closed inelastic string such that bd B is
obtained by wrapping the string around C, pulling it tight at a point and moving
this point around C while keeping the string tight. Conversely, given a planar convex
disc C, each convex disc B obtained in this way is a billiard table with caustic
C. This nice result shows that the planar billiard tables with a given caustic may be
obtained from the caustic by a generalization of the common gardener (see Fig. 10.2)
construction of ellipses.
Lazutkin [631] related the problem of eigenvalues of the Laplace operator on
a planar billiard table B to the existence of convex caustics and showed that, for
billiard tables of class C 553 and with positive curvature, there exists a large family
of caustics. 553 was reduced to 6 by Douady [277]. The problem about the nonexistence
of caustics was studied by Mather [694] and Hubacher [524]. In [422] the
author showed that in the sense of Baire categories, there is only a meagre set of
billiard tables in E 2 which have caustics.
Refining a result of Berger [98], Gruber [433] proved the following result. Its
proof relies on Alexandrov’s differentiability theorem.
Theorem 10.5. Among all convex billiard tables in E d , d ≥ 3, it is only the solid
ellipsoids that have convex caustics. The caustics are precisely the confocal solid
ellipsoids contained in their interiors and, moreover, the intersection of all confocal
ellipsoids.
For further results on caustics of planar billiard tables we refer to the articles of
Gutkin and Katok [458], Knill [603] and Gutkin [457].
B