  Computer simulations of hard spheres and disks are used to estimate the most
probable cavity size, $\xi_{\rm cavity}$, and a ``rattle'' size, $\xi_{\rm
rattle}$, over which a particle can translate holding all other particles
fixed. Both of these measures of free volume appear to extrapolate to zero at
the random close packed density, $\rho_{\rm rcp}$, close to the density where
extrapolations of the viscosity diverge. We also identify the onset of caging
as the density at which cavities cluster. These results suggest that
vitrification in hard core fluids can be viewed as a geometrical phenomenon,
and that {\em local} free volume measures can identify the location of the
onset of liquid-like dynamics, the complex dynamics of caged liquids, and
vitrification.
