  We study dynamical properties of the billiard flow on convex polyhedra away
from a neighbourhood of the non-smooth part of the boundary, called
``pockets''. We prove there are only finitely many immersed periodic tubes
missing the pockets and moreover establish a new quantitative estimate for the
lengths of such tubes. This extends well-known results in dimension $2$. We
then apply these dynamical results to prove a quantitative Laplace
eigenfunction mass concentration near the pockets of convex polyhedral
billiards. As a technical tool for proving our concentration results on
irrational polyhedra, we establish a control-theoretic estimate on a product
space with an almost-periodic boundary condition. This extends previously known
control estimates for periodic boundary conditions, and seems to be of
independent interest.
