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 distributio
n 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.
