  Obtaining general relations between macroscopic properties of random
assemblies, such as density, and the microscopic properties of their
constituent particles, such as shape, is a foundational challenge in the study
of amorphous materials. By leveraging existing understanding of the random
packing of spherical particles, we estimate the random packing density for all
sufficiently spherical shapes. Our method uses the ensemble of random packing
configurations of spheres as a reference point for a perturbative calculation,
which we carry to linear order in the deformation. A fully analytic calculation
shows that all sufficiently spherical shapes pack more densely than spheres.
Additionally, we use simulation data for spheres to calculate numerical
estimates for nonspherical particles and compare these estimates to
simulations.
