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 ($\simeq S^3$) 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$V_{n-2}$denotesthevolumeoftypicalclosed$(n-2)$-sectionofthehorizonand$M$istypicalmassscale,ratherthan$C\lesssim (GM)^{1/(n-2)}$intermsofthehooplength$C$, which holds only when $n=3$.