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The existence of both a minimum mass and a minimum density in nature, in thepresence of a positive cosmological constant, is one of the most intriguingresults in classical general relativity. These results follow rigorously fromthe Buchdahl inequalities in four dimensional de Sitter space. In this work, weobtain the generalized Buchdahl inequalities in arbitrary space-time dimensionswith $\Lambda \neq 0$ and consider both the de Sitter and anti-de Sitter cases.The dependence on $D$, the number of space-time dimensions, of the minimum andmaximum masses for stable spherical objects is explicitly obtained. Theanalysis is then extended to the case of dark energy satisfying an arbitrarylinear barotropic equation of state. The Jeans instability of barotropic darkenergy is also investigated, for arbitrary $D$, in the framework of a simpleNewtonian model with and without viscous dissipation, and we determine thedispersion relation describing the dark energy$-$matter condensation process,along with estimates of the corresponding Jeans mass (and radius). Finally, thequantum mechanical implications of mass limits are investigated, and we showthat the existence of a minimum mass scale naturally leads to a model in whichdark energy is composed of a `sea' of quantum particles, each with an effectivemass proportional to $\Lambda^{1/4}$.

The minimum mass of a spherically symmetric object in $D$-dimensions, and its implications for the mass hierarchy problem