  In this paper we deal with spectral optimization for the Robin Laplacian on a
family of planar domains admitting parallel coordinates, namely a fixed-width
strip built over a smooth closed curve and the exterior of a convex set with a
smooth boundary. We show that if the curve length is kept fixed, the first
eigenvalue referring to the fixed-width strip is for any value of the Robin
parameter maximized by a circular annulus. Furthermore, we prove that the
second eigenvalue in the exterior of a convex domain $\Omega$ corresponding to
a negative Robin parameter does not exceed the analogous quantity for a disk
whose boundary has a curvature larger than or equal to the maximum of that for
$\partial\Omega$.
