  We are concerned with the time-harmonic elastic scattering due to an
inhomogeneous elastic material inclusion located inside a uniformly homogeneous
isotropic medium. We establish a sharp stability estimate of logarithmic type
in determining the support of the elastic scatterer, independent of its
material content, by a single far-field measurement when the support is a
convex polyhedral domain in $\mathbb{R}^n$, $n=2,3$. Our argument in
establishing the stability result is localized around a corner of the medium
scatterer. This enables us to further establish a byproduct result by proving
that if a generic medium scatterer, not necessary to be a polyhedral shape,
possesses a corner, then there exists a positive lower bound of the scattered
far-field patterns. The latter result indicates that if an elastic material
object possesses a corner on its support, then it scatters every incident wave
stably and invisibility phenomenon does not occur.
