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

Two unique Identifiability results for inverse scattering problems within polyhedral geometries