Non-concavity of the Robin ground state

On a convex bounded Euclidean domain, the ground state 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 should have similar concavity properties. The aim of this paper is to show that this is false by analyzing the perturbation problem from the Neumann case. First, we classify all convex polyhedral domains on which the first variation of the ground state with respect to the Robin parameter at zero is not a concave function. Then, we conclude from this that the Robin ground state is not log-concave (and indeed even has some superlevel sets which are non-convex) for small Robin parameter on polyhedral convex domains outside a special class, and hence also on convex domains with smooth boundary which approximate these in Hausdorff distance.