Gravitational equilibrium in the presence of a positive cosmological constant

We reconsider the virial theorem in the presence of a positive cosmological constant \ensuremath{\Lambda}. Assuming steady state, we derive an inequality of the form $\ensuremath{\rho}g~A(\ensuremath{\Lambda}/8\ensuremath{\pi}{G}_{N})$ for the mean density \ensuremath{\rho} of the astrophysical object. The parameter A depends only on the shape of the object. With a minimum at ${A}_{\mathrm{sphere}}=2,$ its value can increase by several orders of magnitude as the shape of the object deviates from a spherically symmetric one. This indicates that flattened matter distributions such as, e.g., clusters or superclusters, with low density, cannot be in gravitational equilibrium.