Summary: This paper considers the onset of convection in shallow cylindrical containers heated from below where the boundary which forms the cross-section of the cylinder is of arbitrary shape. The lateral dimensions of the bonndary are assumed to be much larger than the characteristic wavelength of convection, allowing use of the methods of multiple scales and matched asymptotic expansions. A theory is constructed which describes a local onset of convection consisting of rolls confined to the neighbourhood of the diameter of the boundary (that is, the chord of maximum length that spans the boundary). Use of a Fourier transformation allows the curvature of both the boundary and the resulting roll pattern to be taken into account within a weakly nonlinear framework. Explicit solutions are obtained for the limiting case where the boundary curvature is large. Certain features of the nonlinear development of the roll pattern and the manner in which it spreads to other parts of the domain are discussed with reference to particular geometries. The results suggest that the pattern evolves in a way that maximizes the area of convection. One implication of the theory is that the onset of convection in shallow rectangular containers does not occur in the form of rolls parallel to the shorter sides.