A systematic analysis of the dependence of the confinement loss of an anti-resonant capillary on the curvature of the core surround is presented. The core boundary is described by circular arcs and the construction allows for a wide range of core shapes to be considered. It is found that both negative and positive curvatures substantially reduce the confinement loss relative to that of a circular anti-resonant capillary and that this effect is insensitive to the size of the core relative to the wavelength and to the properties of the glass capillary wall. In contrast, for a solid core surround there is a small increase in the confinement loss with curvature. Results of scalar and vector calculations are shown to be similar. A qualitative explanation of the results is proposed based on azimuthal confinement of the wave fields generated by the curved boundaries.