Gaussian Cooling and O Algorithms for Volume and Gaussian Volume.

We present an O* (n(3)) randomized algorithm for estimating the volume of a wellrounded convex body (e.g., K subset of R-n if Bn subset of K and E-X similar to K(parallel to X parallel to(2)) = O*(n)) given by a membership oracle, improving on the previous best complexity of O* (n(4)). The new algorithmic ingredient is an accelerated cooling schedule where the rate of cooling increases with the temperature. Previously, the known approach for potentially achieving this asymptotic complexity relied on a positive resolution of the Kannan Lovasz Simonovits (KLS) hyperplane conjecture, a central open problem in convex geometry. We also obtain an O*(n(3)) randomized algorithm for integrating a standard Gaussian distribution over an arbitrary convex set containing the unit ball. Both the volume and the Gaussian volume algorithms use an improved algorithm for sampling a Gaussian distribution restricted to a convex body. In this latter setting, as we show, the KLS conjecture holds and for a spherical Gaussian distribution with variance sigma(2), the sampling complexity is O* (max{n(3), sigma(2)n(2)}) for the first sample and O* (max{n(2), sigma(2)n(2)}) for every subsequent sample.