The minimum mass of a spherically symmetric object in $D$-dimensions,
  and its implications for the mass hierarchy problem
  The existence of both a minimum mass and a minimum density in nature, in the
presence of a positive cosmological constant, is one of the most intriguing
results in classical general relativity. These results follow rigorously from
the Buchdahl inequalities in four dimensional de Sitter space. In this work, we
obtain the generalized Buchdahl inequalities in arbitrary space-time dimensions
with $\Lambda \neq 0$ and consider both the de Sitter and anti-de Sitter cases.
The dependence on $D$, the number of space-time dimensions, of the minimum and
maximum masses for stable spherical objects is explicitly obtained. The
analysis is then extended to the case of dark energy satisfying an arbitrary
linear barotropic equation of state. The Jeans instability of barotropic dark
energy is also investigated, for arbitrary $D$, in the framework of a simple
Newtonian model with and without viscous dissipation, and we determine the
dispersion relation describing the dark energy$-$matter condensation process,
along with estimates of the corresponding Jeans mass (and radius). Finally, the
quantum mechanical implications of mass limits are investigated, and we show
that the existence of a minimum mass scale naturally leads to a model in which
dark energy is composed of a `sea' of quantum particles, each with an effective
mass proportional to $\Lambda^{1/4}$.
