Birkhoff billiards are insecure (Q1001667)

Two points \(A, B\) of a Riemannian manifold \(M\) are called \textit{secure} if there exists a finite set of points \(S\subset M\setminus \{A, B\}\) such that every geodesic connecting \(A\) and \(B\) passes through a point of \(S\). The manifold is called \textit{secure} if any pair of its points is secure; for example a flat torus of any dimension is secure. In this note it is proved that a compact plane billiard domain, bounded by a smooth curve, is insecure. An appendix, written by R. Schwartz, contains an useful result on rational approximation.