  We calculate the electronic compressibility arising from electron-electron
interactions for a graphene bilayer within the Hartree-Fock approximation. We
show that, due to the chiral nature of the particles in this system, the
compressibility is rather different from those of either the two-dimensional
electron gas or ordinary semiconductors. We find that an inherent competition
between the contributions coming from intra-band exchange interactions
(dominant at low densities) and inter-band interactions (dominant at moderate
densities) leads to a non-monotonic behavior of the compressibility as a
function of carrier density.
