  We study the numerical computation of an expectation of a bounded function
with respect to a measure given by a non-normalized density on a convex body.
We assume that the density is log-concave, satisfies a variability condition
and is not too narrow. We consider general convex bodies or even the whole
$\R^d$ and show that the integration problem satisfies a refined form of
tractability. The main tools are the hit-and-run algorithm and an error bound
of a multi run Markov chain Monte Carlo method.
