  We prove lower bounds for the worst case error of quadrature formulas that
use given sample points $\X_n = \{ x_1, \dots , x_n \}$. We are mainly
interested in optimal point sets $\X_n$, but also prove lower bounds that hold
with high probability for sets of independently and uniformly distributed
points. As a tool, we use a recent result (and extensions thereof) of Vyb\'iral
on the positive semi-definiteness of certain matrices related to the product
theorem of Schur. The new technique also works for spaces of analytic functions
where known methods based on decomposable kernels cannot be applied.
