In many instances of holographic correspondences between a d dimensional boundary theory and a d+1 dimensional bulk, a simple argument in the boundary theory implies that there must exist a direct relation between the on-shell Euclidean gravitational bulk action and the on-shell Euclidean action of a (d-1)-brane probing the bulk geometry. This relation is crucial for the consistency of holography but puzzling from the bulk perspective. We provide a full bulk derivation in the case of pure gravity. A central role is played by a non-trivial isoperimetric inequality that must be satisfied in a large class of Poincaré-Einstein spaces. Remarkably, this inequality follows from a theorem by John Lee.