It is shown that the "hit-and-run" algorithm for sampling from a convex bod
y K (introduced by R.L. Smith) mixes in time O*(n(2)R(2)/r(2)), where R and
r are the radii of the inscribed and circumscribed balls of K. Thus after
appropriate preprocessing, hit-and-run produces an approximately uniformly
distributed sample point in time O*(n(3)), which matches the best known bou
nd for other sampling algorithms. We show that the bound is best possible i
n terms of R, r and n.