Place a point light source inside a smooth convex billiard table (or mirror). The n-th caustic by reflection is the envelope of the light rays after n reflections. Theorem: each of these caustics, for a generic point light source, has at least 4 cusps. This is a billiard version of "Jacobi’s Last Geometric Statement", concerning the number of cusps of the conjugate locus of a point on a convex surface, proved so far only in the n=1 case. I will show various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory. I will also show computer experiments supporting the conjecture that for an elliptical billiard table and any position of the light source, other than the foci, the n-th caustic by reflection has exactly 4 cusps.
(Joint work with Serge Tabachnikov, part two)
Alain Albouy, Alain Chenciner, Jacques Laskar
