  Eshelby showed that if an inclusion is of elliptic or ellipsoidal shape then
for any uniform elastic loading the field inside the inclusion is uniform. He
then conjectured that the converse is true, i.e., that if the field inside an
inclusion is uniform for all uniform loadings, then the inclusion is of
elliptic or ellipsoidal shape. We call this the weak Eshelby conjecture. In
this paper we prove this conjecture in three dimensions. In two dimensions, a
stronger conjecture, which we call the strong Eshelby conjecture, has been
proved: If the field inside an inclusion is uniform for a single uniform
loading, then the inclusion is of elliptic shape. We give an alternative proof
of Eshelby's conjecture in two dimensions using a hodographic transformation.
As a consequence of the weak Eshelby's conjecture, we prove in two and three
dimensions a conjecture of Polya and Szego on the isoperimetric inequalities
for the polarization tensors. The Polya-Szego conjecture asserts that the
inclusion whose electrical polarization tensor has the minimal trace takes the
shape of a disk or a ball.
