  We consider the inverse conductivity problem in a strictly convex domain
whose boundary is not known. Usually the numerical reconstruction from the
measured current and voltage data is done assuming the domain has a known fixed
geometry. However, in practical applications the geometry of the domain is
usually not known. This introduces an error, and effectively changes the
problem into an anisotropic one. The main result of this paper is a uniqueness
result characterizing the isotropic conductivities on convex domains in terms
of measurements done on a different domain, which we call the model domain, up
to an affine isometry. As data for the inverse problem, we assume the
Robin-to-Neumann map and the contact impedance function on the boundary of the
model domain to be given.
  Also, we present a minimization algorithm based on the use of Cotton--York
tensor, that finds the pushforward of the isotropic conductivity to our model
domain, and also finds the boundary of the original domain up to an affine
isometry. This algorithm works also in dimensions higher than three, but then
the Cotton--York tensor has to replaced with the Weyl--tensor.
