We solve a long-standing open problem posed by Cover and named "The Capacity of the Relay Channel," in the case when the channels from the source to the relay and the destination are binary symmetric channels. Similar to our recent solution of this problem in the Gaussian case, we solve this problem by connecting it to high-dimensional geometry. However, our geometric approach in the binary case significantly deviates from  the Gaussian case, since our treatment of the Gaussian case relied on an extension of the classical isoperimetric inequality on the Euclidean sphere, the counterpart of which does not exist on the Hamming sphere. Instead, we prove a Riesz rearrangement type inequality on the Hamming sphere, which allows us to develop a new upper bound on the capacity of the binary symmetric relay channel. Our argument (and consequently our upper bound) for the binary case is weaker than the one we obtained in the Gaussian case, but nevertheless strong enough to resolve Cover's problem.