“A hole is nothing at all, but it can break your neck.” In a similar fashion to the danger illustrated by this folk paradox, concave regions pose difficulties to theories of visual shape perception. We can readily identify their shapes, but according to principles of how observers determine part boundaries, concavities in a planar surface should have very different figural shapes from the ones that we perceive. In three experiments, we tested the hypothesis that observers perceive local image features differently in simulated 3-D concave and convex regions but use them to arrive at similar shape percepts. Stimuli were shape-from-shading images containing regions that appeared either concave or convex in depth, depending on their orientation in the picture plane. The results show that concavities did not benefit from the same global object-based attention or holistic shape encoding as convexities and that the participants relied on separable spatial dimensions to judge figural shape in concavities. Concavities may exploit a secondary process for shape perception that allows regions composed of perceptually independent features to ultimately be perceived as gestalts.