  Normalizing the Einstein-Hilbert action by the volume functional makes the
theory invariant under constant shifts in the Lagrangian. The associated field
equations then resemble unimodular gravity whose otherwise arbitrary
cosmological constant is now determined as a Machian universal average. We
prove that an empty space-time is necessarily Ricci tensor flat, and
demonstrate the vanishing of the cosmological constant within the scalar field
paradigm. The cosmological analysis, carried out at the mini-superspace level,
reveals a vanishing cosmological constant for a Universe which cannot be closed
as long as gravity is attractive. Finally, we give an example of a normalized
theory of gravity which does give rise to a non-zero cosmological constant.
