Peer-reviewedCC BYInternational Mathematics Research Notices
On the Minkowski inequality near the sphere
- 1Department of Mathematics, Stanford University , Building 380, Stanford, CA 94305, USA
- 2Faculty of Mathematics, University of Vienna , Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
Abstract
We construct a sequence $\{\Sigma _\ell \}_{\ell =1}^\infty $ of closed, axially symmetric surfaces Σ ℓ ⊂ ℝ3 that converges to the unit sphere in W2,p∩ C1 for every p∈ [1,∞ ) and such that, for every $\ell $, & ∫ΣℓHΣℓ−√(16 π |Σℓ|)<0, where HΣ ℓ 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 W2,p∩ C1.
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BibTeX
@article{chodosh2026minkowski,
title = {{On the Minkowski inequality near the sphere}},
author = {Otis Chodosh and Michael Eichmair and Thomas Koerber},
journal = {International Mathematics Research Notices},
year = {2026},
month = jun,
doi = {10.1093/imrn/rnag142},
url = {https://doi.org/10.1093/imrn/rnag142},
note = {CC BY},
}
RIS
TY - JOUR
TI - On the Minkowski inequality near the sphere
AU - Otis Chodosh
AU - Michael Eichmair
AU - Thomas Koerber
PY - 2026
DA - 2026/06/16
JO - International Mathematics Research Notices
DO - 10.1093/imrn/rnag142
UR - https://doi.org/10.1093/imrn/rnag142
AB - We construct a sequence $\{\Sigma _\ell \}_{\ell =1}^\infty $ of closed, axially symmetric surfaces Σ _ℓ ⊂ ℝ^3 that converges to the unit sphere in W^2,p∩ C^1 for every p∈ [1,∞ ) and such that, for every $\ell $, & ∫_Σ_ℓH_Σ_ℓ−√(16 π |Σ_ℓ|)<0, where H_Σ _ℓ 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∩ C^1.
SN - 1073-7928
SN - 1687-0247
ER -