Billiards in convex bodies with acute angles

In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body $K\subset {\mathbb{R}}^d$ has the property that the tangent cone of every non-smooth point $q\in \mathrm{\partial }K$ is acute (in a certain sense) then there is a closed billiard trajectory in $K$.