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On reconstruction in the inverse conductivity problem with one measurement

We consider an inverse problem for electrically conductive material occupying a domain Ω in ℝ2. Let γ be the conductivity of Ω, and D a subdomain of Ω. We assume that γ is a positive constant k on D, k≠1 and is 1 on Ω∖D; both D and k are unknown. The problem is to find a reconstruction formula of D from the Cauchy data on ∂Ω of a non-constant solution u of the equation ∇·γ∇u = 0 in Ω. We prove that if D is known to be a convex polygon such that diamD<dist(D,∂Ω), there are two formulae for calculating the support function of D from the Cauchy data.