Billiards in tubular neighborhoods of manifolds of codimension 1 (Q1969236)

Let \(M\) be a \(C^2\) manifold of codimension one in a Euclidean space, and \(M_{\rho}\) its tubular neighborhood of radius \(\rho\). If \(\rho>0\) is small enough, then every two points in \(\bar{M}_{\rho}\) can be connected by a billiard trajectory that bounces off \(\partial M_{\rho}\). If the manifold \(M\) is \(C^3\) smooth, then the number of bounces can be made less than const\(\cdot\rho^{-1/2}\).