Title: On the Minkowski inequality near the sphere
\nAuthors: Otis Chodosh, Michael Eichmair, Thomas Koerber
\nCategories: math.DG
\nComments: All comments welcome
\n\\\\
\n We construct a sequence $\\{\\Sigma_\\ell\\}_{\\ell=1}^\\infty$ of closed, axially
\nsymmetric surfaces $\\Sigma_\\ell\\subset \\mathbb{R}^3$ that converges to the unit
\nsphere in $W^{2,p}\\cap C^1$ for every $p\\in[1,\\infty)$ and such that, for every
\n$\\ell$, $$
\n\\int_{\\Sigma_{\\ell}}H_{\\Sigma_\\ell}-\\sqrt{16\\,\\pi\\,|\\Sigma_{\\ell}|}<0 $$ where\n$H_{\\Sigma_\\ell}$ is the mean curvature of $\\Sigma_\\ell$. This shows that the\nMinkowski inequality with optimal constant fails even for perturbations of a\nround sphere that are small in $W^{2,p}\\cap C^1$ unless additional convexity\nassumptions are imposed.\n\\\\ ( https:\/\/arxiv.org\/abs\/2306.03848 , 11kb)\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"