An average interpolation is introduced for 3-rectangles and tetrahedra, and optimal order error estimates in the H¹ norm are proved. The constant in the estimate depends “weakly” (improving the results given in Durán (Math. Comp. 68 (1999) 187–199) on the uniformity of the mesh in each direction. For tetrahedra, the constant also depends on the maximum angle of the element. On the other hand, merging several known results (Acosta and Durán, SIAM J. Numer. Anal. 37 (1999) 18–36; Durán, Math. Comp. 68 (1999) 187–199; Krı́zek, SIAM J. Numer. Anal. 29 (1992) 513–520; Al Shenk, Math. Comp. 63 (1994) 105–119), we prove optimal order error for the P ₁-Lagrange interpolation in W^1,p, p>2, with a constant depending on p as well as the maximum angle of the element. Again, under the maximum angle condition, optimal order error estimates are obtained in the H¹ norm for higher degree interpolations.