  We consider the unique determinations of impenetrable obstacles or
diffraction grating profiles in $\mathbb{R}^3$ by a single far-field
measurement within polyhedral geometries. We are particularly interested in the
case that the scattering objects are of impedance type. We derive two new
unique identifiability results for the inverse scattering problem in the
aforementioned two challenging setups. The main technical idea is to exploit
certain quantitative geometric properties of the Laplacian eigenfunctions which
were initiated in our recent works [8,9]. In this paper, we derive novel
geometric properties that generalize and extend the related results in [9],
which further enable us to establish the new unique identifiability results. It
is pointed out that in addition to the shape of the obstacle or the grating
profile, we can simultaneously recover the boundary impedance parameters.
