Hit-and-run mixes fast

It is shown that the “hit-and-run” algorithm for sampling from a convex body K (introduced by R.L. Smith) mixes in time O(nR/r), where R and r are the radii of the inscribed and circumscribed balls of K. Thus after appropriate preprocessing, hit-andrun produces an approximately uniformly distributed sample point in time O(n), which matches the best known bound for other sampling algorithms. We show that the bound is best possible in terms of R, r and n.