The radial deformation of multiwall carbon nanotubes (MWNTs) under hydrostatic pressure is investigated within the continuum elastic approximation. A thin-shell theory, with accurate elastic constants and interwall couplings, allows us to estimate the critical pressure above which the original circular cross-section transforms into radially corrugated ones. The emphasis is placed on the rigorous formulation of the van der Waals interaction between adjacent walls, which we analyze using two different approaches. Possible consequences of the radial corrugation in the physical properties of pressurized MWNTs are also discussed.