Approximating the Volume of General Pfaffian Bodies

We introduce a new powerful method of approximating the volume (and integrals) of vast number of geometric bodies defined by boolean combinations of Pfaffian conditions. The method depends on the polynomial bounds on the VC - Dimensions of the classes of sets to be measured. The resulting approximation algorithms are quite different in spirit from the other up to now known methods, and gives efficient randomized solutions even for such seemingly untouchable problems of statistical physics like computing the volume of sets defined by the ssystems of exponential and polynomial inequalities.

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