  We construct a sequence $\{\Sigma_\ell\}_{\ell=1}^\infty$ of closed, axially
symmetric surfaces $\Sigma_\ell\subset \mathbb{R}^3$ that converges to the unit
sphere in $W^{2,p}\cap C^1$ for every $p\in[1,\infty)$ and such that, for every
$\ell$, $$
\int_{\Sigma_{\ell}}H_{\Sigma_\ell}-\sqrt{16\,\pi\,|\Sigma_{\ell}|}<0 $$ where
$H_{\Sigma_\ell}$ is the mean curvature of $\Sigma_\ell$. This shows that the
Minkowski inequality with optimal constant fails even for perturbations of a
round sphere that are small in $W^{2,p}\cap C^1$ unless additional convexity
assumptions are imposed.
