To date, no large-scale preparative method for arrays of nanotube enables the experimentalist to arbitrarily define changes in the tubes' diameter along their length. To this goal, we start with anodic alumina substrates displaying controlled modulations in pore diameter obtained by alternating "mild" and "hard" electrochemical etching conditions. We then utilize atomic layer deposition (ALD) to coat the internal pore walls with conformal layers of an oxide. Ferromagnetic Fe3O4 tubes of 10 nm wall thickness and 10-30 mu m in length are thus prepared, which replicate the modulated silhouette of the template. Their magnetic properties strongly depend on the presence of diameter modulations. Introducing one or several very short segments of large diameter (150 nm) into an otherwise thin tube (70 nm diameter) brings its initially large coercive field down to a value close to the case of a homogeneously thick tube. Theoretical modeling emphasizes the major influence of the magnetostatic interactions between neighboring tubes. They are enhanced locally at the sites of diameter modulations, which directly translates into a reduction in coercive field.