Alexandrov-Fenchel type inequalities for hypersurfaces in the sphere
Min Chen
TL;DR
This work addresses the open problem of Alexandrov–Fenchel-type inequalities for hypersurfaces in the sphere $\mathbb{S}^{n+1}$ by establishing a sharp relation among quermassintegrals through an inverse curvature flow. The authors construct a monotone quantity $Q_k(t)$ along the flow $X_t = \frac{\nu}{F}$ and prove its decay to zero as the flow converges to the equator, leading to the key inequality $\int_M \sigma_k d\mu_g \ge \sqrt{\eta_k(\mathcal{A}_{k-1})}$ for $0\le k\le n-1$ (with equality only for geodesic spheres). This yields a sharp three-term relation among adjacent quermassintegrals and a non-sharp relation between two adjacent ones, extending the Alexandrov–Fenchel-type framework to spherical geometry. The approach also provides explicit results for $k=1$ and clarifies the role of the monotone quantity $Q_k(t)$ in proving spherical Minkowski-type inequalities via curvature flows.
Abstract
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$.