  We are concerned with the inverse scattering problem of recovering an
inhomogeneous medium by the associated acoustic wave measurement. We prove that
under certain assumptions, a single far-field pattern determines the values of
a perturbation to the refractive index on the corners of its support. These
assumptions are satisfied for example in the low acoustic frequency regime. As
a consequence if the perturbation is piecewise constant with either a
polyhedral nest geometry or a known polyhedral cell geometry, such as a pixel
or voxel array, we establish the injectivity of the perturbation to far-field
map given a fixed incident wave. This is the first unique determinancy result
of its type in the literature, and all of the existing results essentially make
use of infinitely many measurements.
