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. Under certain assumptions, we show that the asymptotic rate of this convergence is at least $$O(1{/}\log \log d)$$O(1/loglogd) in general and $$O(1 / \log d)$$O(1/logd) provided that the semialgebraic set is defined by a single inequality.