We study the property of matter in equilibrium with a static, spherically symmetric black hole in D-dimensional spacetime. It requires this kind of matter has an equation of state (omegaequiv p_r/rho=-1/(1+2kn), k,nin mathbb{N}), which seems to be independent of D. However, when we associate this with specific models, some interesting limits on space could be found: (i)(D=2+2kn) while the black hole is surrounded by cosmic strings; (ii)the black hole can be surrounded by linear dilaton field only in 4-dimensional spacetime. In both cases, D=4 is special.
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