  Let n>2 and let M be an orientable complete finite volume hyperbolic
n-manifold with (possibly empty) geodesic boundary having Riemannian volume
vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston
states that if M has empty boundary then the ratio between vol(M) and ||M|| is
equal to v_n, where v_n is the volume of the regular ideal geodesic n-simplex
in hyperbolic n-space. On the contrary, Jungreis and Kuessner proved that if
the 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 volume
of M is less than k, then the ratio between vol(M) and ||M|| is greater than
v_n-a. As a consequence we show that for every a>0 there exists a compact
orientable hyperbolic n-manifold M with non-empty geodesic boundary such that
the ratio between vol(M) and ||M|| is greater than v_n-a.
  Our argument also works in the case of empty boundary, thus providing a
somewhat new proof of the proportionality principle for non-compact
finite-volume hyperbolic n-manifolds without boundary.
