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A point source of light is placed inside an oval. The $n$-th caustic byreflection is the envelope of the light rays emanating from the light sourceafter $n$ reflections off the curve. We show that each of these caustics, for ageneric point light source, has at least 4 cusps. This is a billiard variationon Jacobi's Last Geometric Statement, concerning the number of cusps of theconjugate locus of a point on a convex surface. We present various proofs,using different ideas, including the curve shortening flow and Legendrian knottheory.

On cusps of caustics by reflection: a billiard variation on Jacobi's Last Geometric Statement