On the Minkowski inequality near the sphere
Year: 2026
Type: article
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 $, $$ \begin{align*} & \int_{\Sigma_{\ell}}H_{\Sigma_\ell}-\sqrt{16\,\pi\,|\Sigma_{\ell}|}<0, \end{align*} $$ where $H_{\Sigma _\ell }$ is the mean curvature of $\Sigma _\ell $. This shows that unless additional convexity assumptions are imposed, the Minkowski inequality with optimal constant fails even for perturbations of a round sphere that are small in $W^{2,p}\cap C^{1}$.
DOI: 10.1093/imrn/rnag142
Authors: Otis Chodosh, Michael Eichmair, Thomas Koerber
Institutions: Stanford University, University of Vienna
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,534
OpenAlex ID: W4379961422