We report evidence demonstrating that a search asymmetry favoring concave over convex targets can be reversed by altering the figure-ground assignment of edges in shapes. Visual search for a concave target among convex distractors is faster than search for a convex target among concave distractors (a search asymmetry). By using shapes with ambiguous local figure-ground relations, we demonstrated that search can be efficient (with search slopes around 10 ms/item) or inefficient (with search slopes around 30-40 ms/item) with the same stimuli, depending on whether edges are assigned to concave or convex "figures." This assignment process can operate in a top-down manner, according to the task set. The results suggest that attention is allocated to spatial regions following the computation of figure-ground relations in parallel across the elements present. This computation can also be modulated by top-down processes.