Alexandrov-Fenchel type inequalities for hypersurfaces in the sphere
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929675158159360 |
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| author | Chen, Min |
| author_facet | Chen, Min |
| contents | The Alexandrov Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, is fundamental in convex geometry. In $\mathbb{R}^{n+1}$, it states: $\int_Mσ_k dμ_g \ge C(n,k)\big(\int_Mσ_{k-1} dμ_g\big)^{\frac{n-k}{n-k+1}}$. In \cite{Brendle-Guan-Li} (see also \cite{Guan-Li-2}), Brendle, Guan, and Li proposed a Conjecture on the corresponding inequalities in $\mathbb{S}^{n+1}$, which implies a sharp relation between two adjacent quermassintegrals: $\mathcal{A}_k(Ω)\ge ξ_{k,k-1}\big(\mathcal{A}_{k-1}(Ω)\big)$, for any $ 1\le k\le n-1$. This is a long-standing open problem. In this paper, we prove a type of corresponding inequalities in $\mathbb{S}^{n+1}:$ $\int_{M}σ_kdμ_g\ge η_k\big(\mathcal{A}_{k-1}(Ω)\big)$ for any $0\le k\le n-1$. This is equivalent to the sharp relation among three adjacent quermassintegrals for hypersurfaces in $\mathbb{S}^{n+1}$(see (\ref{ineq three})), which also implies a non-sharp relation between two adjacent quermassintegrals $\mathcal{A}_{k}(Ω)\ge η_k\big(\mathcal{A}_{k-1}(Ω)\big)$, for any $ 1\le k\le n-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07854 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Alexandrov-Fenchel type inequalities for hypersurfaces in the sphere Chen, Min Differential Geometry Analysis of PDEs The Alexandrov Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, is fundamental in convex geometry. In $\mathbb{R}^{n+1}$, it states: $\int_Mσ_k dμ_g \ge C(n,k)\big(\int_Mσ_{k-1} dμ_g\big)^{\frac{n-k}{n-k+1}}$. In \cite{Brendle-Guan-Li} (see also \cite{Guan-Li-2}), Brendle, Guan, and Li proposed a Conjecture on the corresponding inequalities in $\mathbb{S}^{n+1}$, which implies a sharp relation between two adjacent quermassintegrals: $\mathcal{A}_k(Ω)\ge ξ_{k,k-1}\big(\mathcal{A}_{k-1}(Ω)\big)$, for any $ 1\le k\le n-1$. This is a long-standing open problem. In this paper, we prove a type of corresponding inequalities in $\mathbb{S}^{n+1}:$ $\int_{M}σ_kdμ_g\ge η_k\big(\mathcal{A}_{k-1}(Ω)\big)$ for any $0\le k\le n-1$. This is equivalent to the sharp relation among three adjacent quermassintegrals for hypersurfaces in $\mathbb{S}^{n+1}$(see (\ref{ineq three})), which also implies a non-sharp relation between two adjacent quermassintegrals $\mathcal{A}_{k}(Ω)\ge η_k\big(\mathcal{A}_{k-1}(Ω)\big)$, for any $ 1\le k\le n-1$. |
| title | Alexandrov-Fenchel type inequalities for hypersurfaces in the sphere |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2501.07854 |