Alexandrov-Fenchel type inequalities for hypersurfaces in the sphere

TL;DR

This paper proves Alexandrov-Fenchel type inequalities for hypersurfaces in the sphere, establishing sharp relations among quermassintegrals and addressing a long-standing conjecture in convex geometry.

Contribution

It establishes the first proof of Alexandrov-Fenchel inequalities in the sphere, confirming conjectured relations among quermassintegrals for hypersurfaces.

Findings

Proved inequalities relating integrals of symmetric functions of principal curvatures in spheres.

Established sharp relations among three adjacent quermassintegrals.

Confirmed a long-standing conjecture in convex geometry.

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: $\int_M\sigma_k d\mu_g \ge C(n,k)\big(\int_M\sigma_{k-1} d\mu_g\big)^{\frac{n-k}{n-k+1}}$. 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$.