  Moment-sum-of-squares hierarchies of semidefinite programs can be used to
approximate the volume of a given compact basic semialgebraic set K. The idea
consists of approximating from above the indicator function of K with a
sequence of polynomials of increasing degree d, so that the integrals of these
polynomials generate a convergence sequence of upper bounds on the volume of K.
We show that the asymptotic rate of this convergence is at least O(1/ log log
d).
