Isoperimetric inequality for higher-dimensional black holes

The initial data sets for the five-dimensional Einstein equation have been examined. The system is designed such that the black hole $\left(\simeq S^3\right)$ or the black ring $(\simeq S^2\times S^1)$ can be found. We have found that the typical length of the horizon can become arbitrarily large but the area of characteristic closed two-dimensional submanifold of the horizon is bounded above by the typical mass scale. We conjecture that the isoperimetric inequality for black holes in n-dimensional space is given by $V_{n-2}\lesssim GM,$ where $V_{n-2}$ denotes the volume of a typical closed (n-2)-section of the horizon and M is typical mass scale, rather than $C\lesssim {\mathrm{}\left(\mathrm{GM}\right)\mathrm{}}^{1/\left(n-2\right)}$ in terms of the hoop length C, which holds only when $n=3.\mathrm{}$