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In the framework of gravothermal evolution of an ideal monatomic fluid, I examine the dynamical instability of the fluid sphere in ($N$+1) dimensions by exploiting Chandrasekhar's criterion to each quasistatic equilibrium along the sequence of the evolution. Once the instability is triggered, it would probably collapse into a black hole if no other interaction halts the process. From this viewpoint, the privilege of (3+1)-dimensional spacetime is manifest, as it is the marginal dimensionality in which the ideal monatomic fluid is stable but not too stable. Moreover, it is the unique dimensionality that allows stable hydrostatic equilibrium with positive cosmological constant. While all higher dimensional ($N>3$) spheres are genuinely unstable. In contrast, in (2+1)-dimensional spacetime it is too stable either in the context of Newton's theory of gravity or Einstein's general relativity. It is well known that the role of negative cosmological constant is crucial to have the Bañados-Teitelboim-Zanelli (BTZ) black hole solution and the equilibrium configurations of a fluid disk. Owing to the negativeness of the cosmological constant, there is no unstable configuration for a homogeneous fluid disk to collapse into a naked singularity, which supports the cosmic censorship conjecture. However, BTZ holes of mass $\mathcal{M}_{\rm BTZ}>0$ could emerge from collapsing fluid disks. The implications of spacetime dimensionality are briefly discussed.