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We consider the problem of computing the minimum value $f_{\min,K}$ of apolynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$, which can bereformulated as finding a probability measure $\nu$ on $K$ minimizing $\int_K fd\nu$. Lasserre showed that it suffices to consider such measures of the form$\nu = q\mu$, where $q$ is a sum-of-squares polynomial and $\mu$ is a givenBorel measure supported on $K$. By bounding the degree of $q$ by $2r$ one getsa converging hierarchy of upper bounds $f^{(r)}$ for $f_{\min,K}$. When $K$ isthe hypercube $[-1, 1]^n$, equipped with the Chebyshev measure, the parameters$f^{(r)}$ are known to converge to $f_{\min,K}$ at a rate in $O(1/r^2)$. Weextend this error estimate to a wider class of convex bodies, while alsoallowing for a broader class of reference measures, including the Lebesguemeasure. Our analysis applies to simplices, balls and convex bodies thatlocally look like a ball. In addition, we show an error estimate in $O(\log r /r)$ when $K$ satisfies a minor geometrical condition, and in $O(\log^2 r /r^2)$ when $K$ is a convex body, equipped with the Lebesgue measure. Thisimproves upon the currently best known error estimates in $O(1 / \sqrt{r})$ and$O(1/r)$ for these two respective cases.

Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets