  We give the first rigorous proof of the convergence of Riemannian Hamiltonian
Monte Carlo, a general (and practical) method for sampling Gibbs distributions.
Our analysis shows that the rate of convergence is bounded in terms of natural
smoothness parameters of an associated Riemannian manifold. We then apply the
method with the manifold defined by the log barrier function to the problems of
(1) uniformly sampling a polytope and (2) computing its volume, the latter by
extending Gaussian cooling to the manifold setting. In both cases, the total
number of steps needed is O^{*}(mn^{\frac{2}{3}}), improving the state of the
art. A key ingredient of our analysis is a proof of an analog of the KLS
conjecture for Gibbs distributions over manifolds.
