Several results are presented related to the problem of estimating the complexity M(f//1,. . . , f//N) of the pointwise minimum of n continuous univariate or bivariate functions f//1,. . . , f//N under the assumption that no pair (or triple) of these functions intersect in more than some fixed number s of points. The main result is that in the one-dimensional case M(f//I,. . . , f//N) is the functional inverse of Ackermann's function). In the two-dimensional case, the problem is substantially harder, and the authors have only some initial estimates on M. The treatment of the two-dimensional problem is based on certain properties of the intersection patterns of a collection of planar Jordan curves.