Lateral density structure could exist in the Earth's fluid outer core, owing to the gravitational perturbations induced by the asphericity of the mantle. We take a realistic tomographic image of lateral heterogeneities in the topography of the core-mantle boundary (CMB), as an upper bound for the amplitude of lateral structure in the surface of constant gravitational potential, at the same average depth. With this boundary condition, we apply the theory of Wahr and de Vries [1989>, to estimate quantitatively the corresponding lateral density anomalies in the outer core structure; we find that their value, relative to PREM, must be everywhere smaller than ~0.1%. The resulting, theoretical maps of the outer core are dominated by long-wavelength structure, with a pattern closely resembling that of the assumed CMB topography; they do not correlate with any of the existing tomographic images of the core. ¿ 2001 American Geophysical Union