On the Minkowski inequality near the sphere

We construct a sequence $\{{\mathrm{\Sigma }}_{\mathrm{\ell }}{\}}_{\mathrm{\ell }=1}^{\mathrm{\infty }}$ of closed, axially symmetric surfaces ${\mathrm{\Sigma }}_{\mathrm{\ell }}\subset {\mathbb{R}}^3$ that converges to the unit sphere in $W^{2,p}\cap C^1$ for every $p\in [1,\mathrm{\infty })$ and such that, for every $\mathrm{\ell }$, $${\int }_{{\mathrm{\Sigma }}_{\mathrm{\ell }}}{H}_{{\mathrm{\Sigma }}_{\mathrm{\ell }}}-\sqrt{16\pi \mathrm{\mid }{\mathrm{\Sigma }}_{\mathrm{\ell }}\mathrm{\mid }}<0$$ where $H_{{\mathrm{\Sigma }}_{\mathrm{\ell }}}$ is the mean curvature of ${\mathrm{\Sigma }}_{\mathrm{\ell }}$. 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.