Formulating the problem of determining the communication capacity of channels as a problem in high-dimensional geometry is one of Shannon’s most important insights that has led to the conception of information theory. In his classical paper “Communication in the presence of noise”, 1949, Shannon develops a geometric representation of any point-to-point communication system and provides a geometric proof of the coding theorem for the AWGN channel where the converse is based on a sphere-packing argument in high-dimensional space. We show that a similar geometric approach can be used to prove converses for network communication problems. In particular, we solve a long-standing open problem posed by Cover and named “The Capacity of the Relay Channel,” in Open Problems in Communication and Computation, Springer-Verlag, 1987. The key step in our proof is a strengthening of the isoperimetric inequality on a high-dimensional sphere, which we use to develop a packing argument on a spherical cap, similar to Shannon's original approach. We discuss the promise of this geometric approach for solving other open problems in network information theory.