  In the proof of his systolic inequality, Gromov uses an isometric embedding
of a Riemannian manifold M into the Banach space of bounded functions on M, the
so-called Kuratowski-embedding. Subsequently, it was shown by different authors
that the Kuratowski embedding can be approximated by bi-Lipschitz embeddings
into finite-dimensional Banach spaces. We give a detailed proof for the
existence of such finite-dimensional approximations along the lines suggested
by Larry Guth and go on to discuss quantitative aspects of the problem,
establishing for the dimension of the Banach space a bound which depends on
curvature properties of the manifold.
