Volume of a convex body

Abstract

We arrive at one of the most important applications of the Markov chain Monte Carlo method: the estimation of the volume of a convex body. For a convex body K in low dimensional Euclidean space, say two or three dimensions, it is not too difficult to estimate the volume of K within reasonable relative error using a direct Monte Carlo approach. Depending on how K is presented, it may even be possible to find the volume exactly without too much difficulty. In this chapter, therefore, we imagine the dimension n of the space to be large, and certainly greater than 3.