  We characterize fundamental domains of affine reflection groups as those
polyhedral convex bodies which support a continuous billiard dynamics. We
interpret this characterization in the broader context of Alexandrov geometry
and prove an analogous characterization for isosceles tetrahedra in terms of
continuous quasigeodesic flows. Moreover, we show an optimal regularity result
for convex bodies: the billiard dynamics is continuous if the boundary is of
class $\mathcal{C}^{2,1}$. In particular, billiard trajectories converge to
geodesics on the boundary in this case. Our proof of the latter continuity
statement is based on Alexandrov geometry methods that we discuss resp.
establish first.
