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 (≃S²×S¹) 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≲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≲(GM)^1/(n-2) in terms of the hoop length C, which holds only when n=3.