  We propose an efficient Monte Carlo method for the computation of the volumes
of high-dimensional bodies with arbitrary shape. We start with a region of
known volume within the interior of the manifold and then use the multistate
Bennett acceptance-ratio method to compute the dimensionless free-energy
difference between a series of equilibrium simulations performed within this
object. The method produces results that are in excellent agreement with
thermodynamic integration, as well as a direct estimate of the associated
statistical uncertainties. The histogram method also allows us to directly
obtain an estimate of the interior radial probability density profile, thus
yielding useful insight into the structural properties of such a
high-dimensional body. We illustrate the method by analyzing the effect of
structural disorder on the basins of attraction of mechanically stable packings
of soft repulsive spheres.
