On the Minkowski inequality near the sphere

Abstract

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.

arXiv commentFinal version to appear in Int. Math. Res. Not