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Mean curvatures and symmetry of convex hypersurfaces
Abstract
Let Mn be a C2 closed convex hypersurface in Euclidean space, and σm be its mth mean curvature. We show that M is symmetric with respect to a hyperplane orthogonal to a given direction e, if σm(p)≤σm(q) whenever p−q is parallel to e. For convex hypersurfaces, this settles a conjecture of Li, extends the mean-curvature theorem of Li-Yan-Yao to all σm, and strengthens some earlier results of Li-Nirenberg by removing nondegeneracy assumptions. The proof is based on the theory of mixed volumes, specifically the rigidity of quermassintegrals under Steiner symmetrization.
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BibTeX
@misc{ghomi2026mean,
title = {{Mean curvatures and symmetry of convex hypersurfaces}},
author = {Mohammad Ghomi},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28410},
eprint = {2608.28410},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28410},
}
RIS
TY - UNPB
TI - Mean curvatures and symmetry of convex hypersurfaces
AU - Mohammad Ghomi
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28410
UR - https://arxiv.org/abs/2608.28410
AB - Let M^n be a C^2 closed convex hypersurface in Euclidean space, and σ_m be its mth mean curvature. We show that M is symmetric with respect to a hyperplane orthogonal to a given direction e, if σ_m(p)≤σ_m(q) whenever p−q is parallel to e. For convex hypersurfaces, this settles a conjecture of Li, extends the mean-curvature theorem of Li-Yan-Yao to all σ_m, and strengthens some earlier results of Li-Nirenberg by removing nondegeneracy assumptions. The proof is based on the theory of mixed volumes, specifically the rigidity of quermassintegrals under Steiner symmetrization.
ER -