  We propose new weak error bounds and expansion in dimension one for optimal
quantization-based cubature formula for different classes of functions, such
that piecewise affine functions, Lipschitz convex functions or differentiable
function with piecewise-defined locally Lipschitz or $\alpha$-H\"older
derivatives. This new results rest on the local behaviors of optimal
quantizers, the $L^r$-$L^s$ distribution mismatch problem and Zador's Theorem.
This new expansion supports the definition of a Richardson-Romberg
extrapolation yielding a better rate of convergence for the cubature formula.
An extension of this expansion is then proposed in higher dimension for the
first time. We then propose a novel variance reduction method for Monte Carlo
estimators, based on one dimensional optimal quantizers.
