In the simplest setting $\Omega$ smooth compact and convex in $R^n$ with linear constant speed trajectories that is ($q_t=q_0+t\cdot v$ until the collision point). What is known about the structure and geometry of the of the following set for $t>0$ very small $$C=\{(q,v), |q_t \in \partial\Omega\}$$ And more generally for the sets $$C_n=\{(q,v) | q_t \in \partial \Omega \text{ and there exist exactly }n \, t_1<t_2<…t_n=t \text{ such that }q_{t_i} \in \partial \Omega\}.$$ Any reference would be appreciated. I would like to use a Reynolds transport theorem on $D_n=\{(q,v) | q_t \notin \partial \Omega \text{ and there exist exactly } n\, t_1<t_2<…t_n \text{ such that }q_{t_i} \in \partial \Omega\}.$
Notice that $\partial D_n= C_n \cup C_{n+1}$. This problem must have been encountered before, any reference would be greatly appreciated.
