The Direct Weight Optimization (DWO) approach to estimating a regression function is studied here for the class of approximately linear functions, i.e. functions whose deviation from an affine function is bounded by a known constant. Upper and lower bounds for the asymptotic maximum MSE are given, some of which also hold in the non-asymptotic case and for an arbitrary fixed design. Their coincidence is then studied. Particularly, under mild conditions, it can be shown that there is always an interval in which the DWO-optimal estimator is optimal among all estimators. Experiment design issues are also studied.