On a convex bounded Euclidean domain, the ground state eigenfunction for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state eigenfunctions should have similar concavity properties. In this talk, I show that this is false, by analysing the perturbation problem from the Neumann case. In particular, we prove that on polyhedral convex domains, except in very special cases (which we completely classify) the variation of the ground state with respect to the Robin parameter is not a concave function. We conclude from this that the Robin ground state eigenfunction is not log-concave (and indeed even has some super level sets which are non-convex) for small Robin parameter o polyhedral convex domains outside a special class, and hence also on arbitrary convex domains which approximate these in Hausdorff distance. The results presented in this talk are from my recent paper obtained in joint work with Ben Andrews (ANU in Canberra) and Julie Clutterbuck (Monash in Melbourne).