Volume Estimates and Rapid Mixing

Computing, or at least estimating, the volume of a body is one of the oldest questions in mathematics, studied already in Egypt and continued by Euclid and Archimedes. Here we are mostly concerned with the computational complexity of estimating volumes of convex bodies in R , with n large. The problems and results to be discussed are all rather recent, most of them less than ten years old, and although there was a breakthrough a few years ago and by now there are several substantial and exciting results, there is much to be done. In vague terms, we would like to find a fast algorithm that computes, for each convex body K in R , positive numbers volK and volK such that volK ≤ volK ≤ volK,