  We consider the scattering problem governed by the Helmholtz equation with
inhomogeneity in both `conductivity' in the divergence form and `potential' in
the lower order term. The support of the inhomogeneity is assumed to contain a
convex corner. We prove that, due to the presence of such corner under
appropriate assumptions on the potential and conductivity in the vicinity of
the corner, any incident field scatters. Based on corner scattering analysis we
present a uniqueness result on determination of the polygonal convex hull of
the support of admissible inhomogeneities, from scattering data corresponding
to one single incident wave. These results require only certain regularity
around the corner for the coefficients modeling the inhomogeneity, whereas away
from the corner they can be quite general. Our main results on scattering and
inverse scattering are established for $\mathbb{R}^2$, while some analytic
tools are developed in any dimension $n\geq 2$.
