  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. In this paper we 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 stat 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 arbitrary convex
domains which approximate these in Hausdorff distance.
