Abstract.For a convex body B in R 3 which is invariant under rotations around one coordinate axis and has a smooth boundary of bounded nonzero curvature, the lattice point discrepancy P B (t) (number of integer points minus volume) of a linearly dilated copy √ tB is estimated from below. On the basis of a recent method of K. Soundararajan [16] an Ω -bound is obtained that improves upon all earlier results of this kind.