  We introduce a "Hamiltonian"-like function, called the volume function,
indispensable to describe the ensemble of jammed matter such as granular
materials and emulsions from a geometrical point of view. The volume function
represents the available volume of each particle in the jammed systems. At the
microscopic level, we show that the volume function is the Voronoi volume
associated to each particle and in turn we provide an analytical formula for
the Voronoi volume in terms of the contact network, valid for any dimension. We
then develop a statistical theory for the probability distribution of the
volumes in 3d to calculate an average volume function coarse-grained at a
mesoscopic level. The salient result is the discovery of a mesoscopic volume
function inversely proportional to the coordination number. Our analysis is the
first step toward the calculation of macroscopic observables and equations of
state using the statistical mechanics of jammed matter, when supplemented by
the condition of mechanical equilibrium of jamming that properly defines jammed
matter at the ensemble level.
