To solve the ill-posed problem of shape-from-shading, the visual system often relies on prior assumptions such as illumination from above or viewpoint from above. Here we demonstrate that a third prior assumption is used--namely that the surface is globally convex. We use complex surface shapes that are realistically rendered with computer graphics, and we find that performance in a local-shape-discrimination task is significantly higher when the shapes are globally convex than when they are globally concave. The results are surprising because the qualitative global shapes of the surfaces are perceptually unambiguous. The results generalise findings such as the hollow-potato illusion (Hill and Bruce 1994 Perception 23 1335-1337) which consider global shape perception only.