The inverse conductivity problem with one measurement: uniqueness for convex polyhedra

Let denote a smooth domain in containing the closure of a convex polyhedron D. Set equal to the characteristic function of D. We find a flux g so that if u is the nonconstant solution of <!-- MATH $\operatorname{div}\;((1 + {\chi _D})\nabla u) = 0$ --> in with <!-- MATH $\frac{{\partial u}}{{\partial n}} = g$ --> on <!-- MATH $\partial \Omega$ --> , then D is uniquely determined by the Cauchy data g and <!-- MATH $f \equiv u/\partial \Omega$ --> .