  Almost four decades ago, Bergman and Milton independently showed that the
isotropic effective electric permittivity of a two-phase composite material
with a given volume fraction is constrained to lie within lens-shaped regions
in the complex plane that are bounded by two circular arcs. An implication of
particular significance is a set of limits to the maximum and minimum
absorption of an isotropic composite material at a given frequency. Here, after
giving a short summary of the underlying theory, we show that the bound
corresponding to one of the circular arcs is at least almost optimal by
introducing a certain class of hierarchical laminates. In regard to the second
arc, we show that a tighter bound can be derived using variational methods.
This tighter bound is optimal as it corresponds to assemblages of doubly coated
spheres, which can be easily approximated by more realistic microstructures. We
briefly discuss the implications for related problems, including bounds on the
complex polarizability.
