  We prove the curse of dimensionality in the worst case setting for
multivariate numerical integration for various classes of smooth functions. We
prove the results when the domains are isotropic convex bodies with small
diameter satisfying a universal $\psi_2$-estimate. In particular, we obtain the
result for the important class of volume-normalized $\ell_p^d$-balls in the
complete regime $2\leq p \leq \infty$. This extends a result in a work of A.
Hinrichs, E. Novak, M. Ullrich and H. Wo\'zniakowski [J. Complexity, 30(2),
117-143, 2014] to the whole range $2\leq p \leq \infty$, and additionally
provides a unified approach. The key ingredient in the proof is a deep result
from the theory of Asymptotic Geometric Analysis, the thin-shell volume
concentration estimate due to O. Gu\'edon and E. Milman. The connection of
Asymptotic Geometric Analysis and Information-based Complexity revealed in this
work seems promising and is of independent interest.
