  In this note, I derive the Chandrasekhar instability of a fluid sphere in
($N$+1)-dimensional Schwarzschild-Tangherlini spacetime and take the
homogeneous (uniform energy density) solution for illustration. Qualitatively,
the effect of positive (negative) cosmological constant tends to destabilize
(stabilize) the sphere. In the absence of cosmological constant, the privileged
position of (3+1)-dimensional spacetime is manifest in its own right. As it is
the marginal dimensionality in which a monatomic ideal fluid sphere is stable
but not too stable to trigger the onset of gravitational collapse. Furthermore,
it is the unique dimensionality that can accommodate stable hydrostatic
equilibrium with positive cosmological constant. However, given the current
cosmological constant observed no stable configuration can be larger than
$10^{21}~{\rm M}_\odot$. On the other hand, in (2+1) dimensions it is too
stable either in the context of Newtonian Gravity (NG) or Einstein's General
Relativity (GR). In GR, the role of negative cosmological constant is crucial
not only to guarantee fluid equilibrium (decreasing monotonicity of pressure)
but also to have the Ba{\~n}ados-Teitelboim-Zanelli (BTZ) solution. Owing to
the negativeness of the cosmological constant, there is no unstable
configuration for a homogeneous fluid disk with mass $0<\mathcal{M}\leq0.5$ to
collapse into a naked singularity, which supports the Cosmic Censorship
Conjecture. However, the relativistic instability can be triggered for a
homogeneous disk with mass $0.5<\mathcal{M}\lesssim0.518$ under causal limit,
which implies that BTZ holes of mass $\mathcal{M}_{\rm BTZ}>0$ could emerge
from collapsing fluid disks under proper conditions. The implicit assumptions
and implications are also discussed.
