  We give a complete solution to an open problem of Thomas Cover in 1987 about
the capacity of a relay channel in the general discrete memoryless setting
without any additional assumptions. The key step in our approach is to lower
bound a certain soft-max of a stochastic process by convex geometry methods,
which is based on two ideas: First, the soft-max is lower bounded in terms of
the supremum of another process, by approximating a convex set with a polytope
with bounded number of vertices. Second, using a result of Pajor, the supremum
of the process is lower bounded in terms of packing numbers by means of
mixed-volume inequalities (Minkowski's first inequality).
