Shape identification in inverse medium scattering problems with a single far-field pattern

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.