Let n ≥ 3, let M be an orientable complete finite-volume hyperbolic n–manifold with
compact (possibly empty) geodesic boundary, and let Vol(M) and ∥M∥ be the
Riemannian volume and the simplicial volume of M. A celebrated result
by Gromov and Thurston states that if ∂M = ∅ then Vol(M) ∕ ∥M∥ = vn,
where vn is the volume of the regular ideal geodesic n–simplex in hyperbolic
n–space. On the contrary, Jungreis and Kuessner proved that if ∂M≠∅ then
Vol(M) ∕ ∥M∥ < vn.
We prove here that for every η > 0 there exists k > 0 (only depending on η
and n) such that if Vol(∂M) ∕ Vol(M) ≤ k, then Vol(M) ∕ ∥M∥≥ vn − η. As a
consequence we show that for every η > 0 there exists a compact orientable
hyperbolic n–manifold M with nonempty geodesic boundary such that
Vol(M) ∕ ∥M∥≥ vn − η.
Our argument also works in the case of empty boundary, thus providing a
somewhat new proof of the proportionality principle for noncompact finite-volume
hyperbolic n–manifolds without geodesic boundary.