  In this paper, we establish two sharp quantitative results for the direct and
inverse time-harmonic acoustic wave scattering. The first one is concerned with
the recovery of the support of an inhomogeneous medium, independent of its
contents, by a single far-field measurement. For this challenging inverse
scattering problem, we establish a sharp stability estimate of logarithmic type
when the medium support is a polyhedral domain in $\mathbb{R}^n$, $n=2,3$. The
second one is concerned with the stability for corner scattering. More
precisely if an inhomogeneous scatterer, whose support has a corner, is probed
by an incident plane-wave, we show that the energy of the scattered far-field
possesses a positive lower bound depending only on the geometry of the corner
and bounds on the refractive index of the medium there. This implies the
impossibility of approximate invisibility cloaking by a device containing a
corner and made of isotropic material. Our results sharply quantify the
qualitative corner scattering results in the literature, and the corresponding
proofs involve much more subtle analysis and technical arguments. As a
significant byproduct of this study, we establish a quantitative Rellich's
theorem that continues smallness of the wave field from the far-field up to the
interior of the inhomogeneity. The result is of significant mathematical
interest for its own sake and is surprisingly not yet known in the literature.
