We show how simple 1-D geometrical calculations (but along all maximal segments of the parameter or control set !) can be used to establish the wellposedness of a non-linear least-square (NLLS) problem and the absence of local minima in the corresponding error function. These sufficient conditions, which are shown to be sharp by elementary examples, are based on the use of the recently developped "size x curvature" conditions for prooving that the output set is strictly quasiconvex. The use of this geometrical theory as a numerical or theoretical tool is discussed. Finally, application to regularized NLLS problem is shown to give new information on the choice of the regularizing parameter.