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In the framework of gravothermal evolution of an ideal monatomic fluid, Iexamine the dynamical instability of the fluid sphere in ($N$+1) dimensions byexploiting Chandrasekhar's criterion to each quasistatic equilibrium along thesequence of the evolution. Once the instability is triggered, it would probablycollapse into a black hole if no other interaction halts the process. From thisviewpoint, the privilege of (3+1)-dimensional spacetime is manifest, as it isthe marginal dimensionality in which the ideal monatomic fluid is stable butnot too stable. Moreover, it is the unique dimensionality that allows stablehydrostatic equilibrium with positive cosmological constant. While all higherdimensional ($N>3$) spheres are genuinely unstable. In contrast, in(2+1)-dimensional spacetime it is too stable either in the context of Newton'stheory of gravity or Einstein's general relativity. It is well known that therole of negative cosmological constant is crucial to have theBa\~nados-Teitelboim-Zanelli (BTZ) black hole solution and the equilibriumconfigurations of a fluid disk. Owing to the negativeness of the cosmologicalconstant, there is no unstable configuration for a homogeneous fluid disk tocollapse into a naked singularity, which supports the cosmic censorshipconjecture. However, BTZ holes of mass $\mathcal{M}_{\rm BTZ}>0$ could emergefrom collapsing fluid disks. The implications of spacetime dimensionality arebriefly discussed.

Gravothermal Phase Transition, Black Holes and Space Dimensionality