J. Aust. Math. Soc.  80 (2006), 375-382
The double cover relative to a convex domain and the relative isoperimetric inequality

Jaigyoung Choe
  Department of Mathematics
  Seoul National University
  Seoul 151-742

We prove that a domain $\Omega$ in the exterior of a convex domain $C$ in a four-dimensional simply connected Riemannian manifold of nonpositive sectional curvature satisfies the relative isoperimetric inequality $64\pi^2 \operatorname{Vol}(\Omega)^3  \leq \operatorname{Vol}(\partial \Omega \sim \partial C)^4$. Equality holds if and only if $\Omega$ is an Euclidean half ball and $\partial \Omega \sim \partial C$ is a hemisphere.
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