Abstract

Given a high dimensional convex body K ` IR n by a separation oracle, we can approximate its volume with relative error ", using O (n 5 ) oracle calls. Our algorithm also brings the body into isotropic position. As all previous randomized volume algorithms, we use "rounding" followed by a multiphase Monte-Carlo (product estimator) technique. Both parts rely on sampling (generating random points in K), which is done by random walk. Our algorithm introduces three new ideas: ffl the use of the isotropic position (or at least an approximation of it) for rounding, ffl the separation of global obstructions (diameter) and local obstructions (boundary problems) for fast mixing, and ffl a stepwise interlacing of rounding and sampling. 1 . Introduction For a variety of geometric objects, classical results characterize various geometric parameters. Many of these results are useful even in practical situations: they can easily be transformed into efficient algorithms. Some other theorem...

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