This letter studies the time-optimal control problem for a chain of multidimensional integrators subject to convex state and input constraints. It is shown that, by performing a suitable change of variables, this problem can be cast as a single convex program for certain realizations of the boundary conditions. For general realizations, the problem is nonconvex. A constructive procedure based on the solution of a sequence of convex problems is proposed in order to approach asymptotically a global optimum. The proposed approach is demonstrated on three applications involving aerial and space vehicle systems.