  We show that there are shape-independent upper bounds to the extinction cross
section per unit volume of randomly oriented nanoparticles, given only material
permittivity. Underlying the limits are restrictive sum rules that constrain
the distribution of quasistatic eigenvalues. Surprisingly, optimally-designed
spheroids, with only a single quasistatic degree of freedom, reach the upper
bounds for four permittivity values. Away from these permittivities, we
demonstrate computationally-optimized structures that surpass spheroids and
approach the fundamental limits.
