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We prove that a region  D  outside a convex set  C  with smooth boundary in a three-dimensional Cartan-Hadamard manifold  M  satisfies the relative isoperimetric inequality area    , with equality if and only if  D  is isometric to a Euclidean half ball. We also prove similar sharp inequalities when the sectional curvature of  M  is bounded above by a negative constant.

The relative isoperimetric inequality in Cartan-Hadamard 3-manifolds