  We study a metric version of the simplicial volume on Riemannian manifolds,
the Lipschitz simplicial volume, with applications to degree theorems in mind.
We establish a proportionality principle and a product inequality from which we
derive an extension of Gromov's volume comparison theorem to products of
negatively curved manifolds or locally symmetric spaces of non-compact type. In
contrast, we provide vanishing results for the ordinary simplicial volume; for
instance, we show that the ordinary simplicial volume of non-compact locally
symmetric spaces with finite volume of Q-rank at least 3 is zero.
