Gravitational equilibrium in the presence of a positive cosmological constant

We reconsider the virial theorem in the presence of a positive cosmological constant Lambda. Assuming steady state, we derive an inequality of the form rhogreater than or equal toA(Lambda/8piG(N)) for the mean density rho of the astrophysical object. The parameter A depends only on the shape of the object. With a minimum at A(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.