  We consider a hierarchy of upper approximations for the minimization of a
polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$ proposed recently
by Lasserre (arXiv:1907.097784, 2019). This hierarchy relies on using the
push-forward measure of the Lebesgue measure on $K$ by the polynomial $f$ and
involves univariate sums of squares of polynomials with growing degrees $2r$.
Hence it is weaker, but cheaper to compute, than an earlier hierarchy by
Lasserre (SIAM Journal on Optimization 21(3), 864--885, 2011), which uses
multivariate sums of squares. We show that this new hierarchy converges to the
global minimum of $f$ at a rate in $O(\log^2 r / r^2)$ whenever $K$ satisfies a
mild geometric condition, which holds, e.g., for convex bodies and for compact
semialgebraic sets with dense interior. As an application this rate of
convergence also applies to the stronger hierarchy based on multivariate sums
of squares, which improves and extends earlier convergence results to a wider
class of compact sets. Furthermore, we show that our analysis is near-optimal
by proving a lower bound on the convergence rate in $\Omega(1/r^2)$ for a class
of polynomials on $K=[-1,1]$, obtained by exploiting a connection to orthogonal
polynomials.
