Experiments on elastomers have shown that triaxial tensions can induce a material to exhibit holes that were not previously evident. Analytic work in nonlinear elasticity has established that such cavity formation may indeed be an elastic phenomenon: sufficiently large prescribed boundary deformations yield a hole-creating deformation as the energy minimizer whenever the elastic energy is of slow growth. In this lecture the speaker will discuss the use of isoperimetric arguments to establish that a radial deformation, producing a spherical cavity, is the energy minimizer in a general class of isochoric deformations that are discontinuous at the center of a ball and produce a (possibly non-symmetric) cavity in the deformed body. The key ingredient is a new radial-symmetrization procedure that is appropriate for problems where the symmetrized mapping must be one-to-one in order to prevent interpenetration of matter.