  This paper is devoted to the analysis of a second order method for recovering
the \emph{a priori} unknown shape of an inclusion $\omega$ inside a body
$\Omega$ from boundary measurement. This inverse problem - known as electrical
impedance tomography - has many important practical applications and hence has
focussed much attention during the last years. However, to our best knowledge,
no work has yet considered a second order approach for this problem. This paper
aims to fill that void: we investigate the existence of second order derivative
of the state $u$ with respect to perturbations of the shape of the interface
$\partial\omega$, then we choose a cost function in order to recover the
geometry of $\partial \omega$ and derive the expression of the derivatives
needed to implement the corresponding Newton method. We then investigate the
stability of the process and explain why this inverse problem is severely
ill-posed by proving the compactness of the Hessian at the global minimizer.
