We derive a general closed expression for the local pressure exerted onto the corrugated walls of a channel confining a fluid medium. When the fluid medium is at equilibrium, the local pressure is a functional of the shape of the walls. It is shown that, due to the intrinsic nonlocal character of the interactions among the particles forming the fluid, the applicability of approximate schemes such as the concept of a surface of tension or morphometric thermodynamics is limited to wall curvatures that are small compared to the range of particle-particle interactions.