  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.
