We study dynamical properties of the billiard flow on $3$ dimensional convex
polyhedrons away from "pockets" and establish a finite tube condition for
rational polyhedrons that extends well-known results in dimension $2$.
Furthermore, we establish a new quantitative estimate for lengths of periodic
tubes in irrational polyhedrons. 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, we establish a control-theoretic estimate on a product space with an
almost-periodic boundary condition that extends previously known estimates for
periodic boundary conditions, which we believe should be of independent
interest.