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 (≅S3) or the black ring (≅S2xS1) 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 Vn-2 < or approx. GM, where Vn-2 denotes the volume of a typical closed (n-2)-section of the horizon and M is typical mass scale, rather than C < or approx. (GM)1/(n-2) in terms of the hoop length C, which holds only when n=3