Alexandrov--Fenchel inequalities for closed axisymmetric hypersurfaces
Abstract
We establish sharp Alexandrov--Fenchel inequalities for closed, embedded, axisymmetric hypersurfaces Sigma in n+1 dimensional Euclidean space R^{n+1}, the dimension n > or = 2, with admissible C^{1,1} profiles. Let H_j be the j-th normalized mean curvature. If Sigma is axiconvex, i.e., every hyperplane orthogonal to the symmetry axis meets the enclosed domain in an n-dimensional ball or empty, then ((∫ΣHℓ dA)/|𝕊n|)1/(n−ℓ) ≤ ((∫ΣHk dA)/|𝕊n|)1/(n−k), 0≤ℓ<k≤ n−1. Without axiconvexity, the inequalities hold for sign-corrected integrals ∫Σεθ jHj dA, where varepsilon_{theta}= {sgn}(sin {theta}) denotes the sign of the azimuthal principal curvatures. As applications, the Alexandrov--Fenchel inequalities remain valid whenever both l and k are even. Equality in each of the above inequalities holds if and only if Sigma is a round sphere. The proof adapts the rearrangement and tangent angle folding approach of Dalphin-Henrot-Masnou-Takahashi from surfaces of revolution in R^3 to n-dimensional axisymmetric hypersurfaces in R^{n+1}. The higher-dimensional construction uses weighted monotone rearrangement to produce convex comparison bodies with C^{1,1} boundaries.
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BibTeX
@misc{sheng2026alexandrovfenchel,
title = {{Alexandrov--Fenchel inequalities for closed axisymmetric hypersurfaces}},
author = {Weimin Sheng and Yinhang Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15311},
eprint = {2610.15311},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15311},
}
RIS
TY - UNPB
TI - Alexandrov--Fenchel inequalities for closed axisymmetric hypersurfaces
AU - Weimin Sheng
AU - Yinhang Wang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15311
UR - https://arxiv.org/abs/2610.15311
AB - We establish sharp Alexandrov--Fenchel inequalities for closed, embedded, axisymmetric hypersurfaces Sigma in n+1 dimensional Euclidean space R^{n+1}, the dimension n > or = 2, with admissible C^{1,1} profiles. Let H_j be the j-th normalized mean curvature. If Sigma is axiconvex, i.e., every hyperplane orthogonal to the symmetry axis meets the enclosed domain in an n-dimensional ball or empty, then ((∫_ΣH_ℓ dA)/|𝕊^n|)^1/(n−ℓ) ≤ ((∫_ΣH_k dA)/|𝕊^n|)^1/(n−k), 0≤ℓ<k≤ n−1. Without axiconvexity, the inequalities hold for sign-corrected integrals ∫_Σε_θ^ jH_j dA, where varepsilon_{theta}= {sgn}(sin {theta}) denotes the sign of the azimuthal principal curvatures. As applications, the Alexandrov--Fenchel inequalities remain valid whenever both l and k are even. Equality in each of the above inequalities holds if and only if Sigma is a round sphere. The proof adapts the rearrangement and tangent angle folding approach of Dalphin-Henrot-Masnou-Takahashi from surfaces of revolution in R^3 to n-dimensional axisymmetric hypersurfaces in R^{n+1}. The higher-dimensional construction uses weighted monotone rearrangement to produce convex comparison bodies with C^{1,1} boundaries.
ER -