The cut-set bound developed by Cover and El Gamal in 1979 has since remained the best known upper bound on the capacity of the Gaussian relay channel. We develop a new upper bound on the capacity of the Gaussian primitive relay channel, which is tighter than the cut-set bound. Our proof uses Gaussian measure concentration to establish geometric relations, satisfied with high probability, between the n-letter random variables associated with a reliable code for communicating over this channel. We then translate these geometric relations into new information inequalities that cannot be obtained with classical methods. Combined with a tensorization argument proposed by Courtade and Ozgur in 2015, our result also implies that the current capacity approximations for Gaussian relay networks, which have linear gap to the cut-set bound in the number of nodes, are order-optimal and lead to a lower bound on the pre-constant.