  (Electric) polarization tensors describe part of the leading order term of
asymptotic voltage perturbations caused by low volume fraction inhomogeneities
of the electrical properties of a medium. They depend on the geometry of the
support of the inhomogeneities and on their admittivity contrast. Corresponding
asymptotic formulas are of particular interest in the design of reconstruction
algorithms for determining the locations and the material properties of
inhomogeneities inside a body from measurements of current flows and associated
voltage potentials on the body's surface. In this work we consider the
two-dimensional case only and provide an analytic representation of the
polarization tensor in terms of spectral properties of the double layer
integral operator associated with the support of simply connected conductivity
inhomogeneities. Furthermore, we establish that an (infinitesimal) simply
connected inhomogeneity has the shape of an ellipse, if and only if the
polarization tensor is a rational function of the admittivity contrast with at
most two poles whose residues satisfy a certain algebraic constraint. We also
use the analytic representation to provide a proof of the so-called
Hashin-Shtrikman bounds for polarization tensors; a similar approach has been
taken previously by Golden and Papanicolaou and Kohn and Milton in the context
of anisotropic composite materials.
