  Consider time-harmonic acoustic scattering from a bounded penetrable obstacle
$D\subset \mathbb R^N$ embedded in a homogeneous background medium. The index
of refraction characterizing the material inside $D$ is supposed to be H\"older
continuous near the corners. If $D\subset \mathbb R^2$ is a convex polygon, we
prove that its shape and location can be uniquely determined by the far-field
pattern incited by a single incident wave at a fixed frequency. In dimensions
$N \geq 3$, the uniqueness applies to penetrable scatterers of rectangular type
with additional assumptions on the smoothness of the contrast. Our arguments
are motivated by recent studies on the absence of non-scattering wavenumbers in
domains with corners. As a byproduct, we show that the smoothness conditions in
previous corner scattering results are only required near the corners.
