A sharp isoperimetric inequality for rank one symmetric spaces (Q1396521)

Let \(X\) be a rank one symmetric space and \(\Omega\subset X\) a regular domain, that is, a compact domain such that \(\partial\Omega\) is a \(C^\infty\) type surface and if \(X\) is of non-compact type an extra condition on certain mean curvatures of \(\partial\omega\) is imposed. In this paper the author obtains an isoperimetric inequality for regular domains in \(X\), which is sharp for geodesic balls (it has been conjectured that if \(X\) is of non-compact type, geodesic balls are the solutions of the isoperimetric problem). More precisely, in theorem 2, the author shows that for regular domains \(\Omega\) in \(X\) \[ \text{Vol\,}\Omega\leq \int_{\partial\Omega} {\text{Vol(Br)}\over \text{Area}(\partial B_r)}\,dA, \] where \(r: \partial\Omega\to (0,\infty)\) is a function of some mean curvatures, with equality if and only if \(\Omega\) is a geodesic ball.