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Let n>2 and let M be an orientable complete finite volume hyperbolicn-manifold with (possibly empty) geodesic boundary having Riemannian volumevol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurstonstates that if M has empty boundary then the ratio between vol(M) and ||M|| isequal to v_n, where v_n is the volume of the regular ideal geodesic n-simplexin hyperbolic n-space. On the contrary, Jungreis and Kuessner proved that ifthe boundary of M is non-empty, then such a ratio is strictly less than v_n.  We prove here that for every a>0 there exists k>0 (only depending on a and n)such that if the ratio between the volume of the boundary of M and the volumeof M is less than k, then the ratio between vol(M) and ||M|| is greater thanv_n-a. As a consequence we show that for every a>0 there exists a compactorientable hyperbolic n-manifold M with non-empty geodesic boundary such thatthe ratio between vol(M) and ||M|| is greater than v_n-a.  Our argument also works in the case of empty boundary, thus providing asomewhat new proof of the proportionality principle for non-compactfinite-volume hyperbolic n-manifolds without boundary.

The simplicial volume of hyperbolic manifolds with geodesic boundary