  We consider the possibility that the Eddington-Born-Infeld (EBI) modified
gravity provides an alternative explanation for the mass discrepancy in
clusters of galaxies. For this purpose we derive the modified Einstein field
equations, finding an additional "geometrical mass" term which provides an
effective contribution to the gravitational binding energy. Using some
approximations and assumptions for weak gravitational fields, and taking into
account the collisionless relativistic Boltzmann equation, we derive a
generalized version of the virial theorem in the framework of EBI gravity. We
show that the "geometrical mass" term may account for the well known virial
mass discrepancy in clusters of galaxies. We also derive the velocity
dispersion relation for galaxies in the clusters, which could provide an
efficient method for testing EBI gravity from astrophysical observations.
