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  1. ; ; (, Journal of the London Mathematical Society)
    Abstract This paper is concerned with the billiard version of Jacobi's last geometric statement and its generalizations. Given a non‐focal point inside an elliptic billiard table, one considers the family of rays emanating from and the caustic of the reflected family after reflections off the ellipse, for each positive integer . It is known that has at least four cusps and it has been conjectured that it has exactly four (ordinary) cusps. The present paper presents a proof of this conjecture in the special case when the ellipse is a circle. In the case of an arbitrary ellipse, we give an explicit description of the location of four of the cusps of , though we do not prove that these are the only cusps. 
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  2. (, Theoretical and Applied Mechanics)
    Given a projective Finsler metric in a convex domain in the projective plane, that is, a metric in which geodesics are straight lines, consider the respective Finsler billiard system. Choose a generic point inside the table and consider the billiard trajectories that start at this point and undergo N reflection off the boundary. The envelope of the resulting 1-parameter family of straight lines is the Nth caustic by reflection. We prove that, for every N, it has at least four cusps, generalizing a similar result for Euclidean metric, obtained recently jointly with G. Bor. 
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