Add to ORKGNew bounds on the elastic and electric polarization tensors are found for grains of arbitrary shape or connectivity. For a grain shape specified by the characteristic function χ(x), the bounds are given explicitly in terms of the geometric function | \ ̂gc(k)|2. For electric polarizations one of the bounds may be interpreted as the polarization of a homogeneous ellipsoidal inclusion with axes determined by |x(k)|2- The other bound corresponds to a convex sum of polarization tensors for plate-like inclusions. Here the plate normals and weights are specified by | \ ̂gc(k)|2. These bounds are used to predict the range of effective transport properties for hierarchical random suspensions and aggregates that realize the Effective Medium Approxim...