  In many instances of holographic correspondences between a d dimensional
boundary theory and a d+1 dimensional bulk, a direct argument in the boundary
theory implies that there must exist a simple and precise relation between the
Euclidean on-shell action of a (d-1)-brane probing the bulk geometry and the
Euclidean gravitational bulk action. This relation is crucial for the
consistency of holography, yet it is non-trivial from the bulk perspective. In
particular, we show that it relies on a nice isoperimetric inequality that must
be satisfied in a large class of Poincar\'e-Einstein spaces. Remarkably, this
inequality follows from theorems by Lee and Wang.
