On the finite dimensional approximation of the Kuratowski-embedding for compact manifolds

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.