Gravothermal phase transition, black holes and space dimensionality

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