# Convergence rates of moment-sum-of-squares hierarchies for volume approximation of semialgebraic sets

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)\$.