Minoration via mixed volumes and Cover’s problem for general channels

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