The mean sputter yield produced by the impact of a single ion depends on the radii of curvature of the target surface at the point of impact. Using the Sigmund model of ion sputtering, we develop analytical formulas for this dependence for the case in which the radii of curvature are large compared to the size of the ion-induced collision cascade; both locally perpendicular and oblique ion impact are considered. The sputter yield is increased for impact on convex surfaces. The influence of surface curvature along the incident-ion azimuth and perpendicular to it are discussed separately. Our analytical results are in good agreement with Monte Carlo simulations for the specific case of 20 keV Ar ion impact on a cylindrical nanowire consisting of amorphous silicon. We also extend the results for this case to small radii of curvature using both Monte Carlo and molecular dynamics simulations.