Alexandrov-Fenchel type inequalities for hypersurfaces in the sphere

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: $\int_M\sigma_k d\mu_g \ge C(n,k)\big(\int_M\sigma_{k-1} d\mu_g\big)^{\frac{n-k}{n-k+1}}$. 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: $\mathcal{A}_k(\Omega)\ge \xi_{k,k-1}\big(\mathcal{A}_{k-1}(\Omega)\big)$, 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 $\mathbb{S}^{n+1}:$ $\int_{M}\sigma_kd\mu_g\ge \eta_k\big(\mathcal{A}_{k-1}(\Omega)\big)$ 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 $\mathcal{A}_{k}(\Omega)\ge \eta_k\big(\mathcal{A}_{k-1}(\Omega)\big)$, for any $1\le k\le n-1$.