This paper gives sharp rates of convergence for natural versions of the Metropolis algorithm for sampling from the uniform distribution on a convex polytope. The singular proposal distribution , based on a walk moving locally in one of a fixed, finite set of directions, needs some new tools. We get useful bounds on the spectrum and eigenfunctions using Nash and Weyl-type inequalities. The top eigenvalues of the Markov chain are closely related to the Neuman eigenvalues of the polytope for a novel Laplacian.