The relative Alexandrov-Fenchel type inequalities

Abstract

The Alexandrov-Fenchel inequality is one of the important inequalities in convex geometry, and its special form can be regarded as an isoperimetric inequality between higher-order quermassintegrals. In recent years, hypersurfaces with free or capillary boundaries have received much attention. In this paper, we will study a class of relative Alexandrov-Fenchel-type inequalities on convex hypersurfaces with free or capillary boundaries in the unit ball or in half-space from the perspective of differential geometry, and review its recent progress. First, we introduce the higher-order Minkowski-type integral formula for capillary hypersurfaces in the unit ball or half-space. From the first variational point of view, we give the definition of the quermassintegrals of the capillary hypersurfaces in the unit ball or half-space. After that, we use the Minkowski-type formula to construct a locally constrained curvature flow. By analyzing the long-time existence and convergence of the flow, we prove a class of relative Alexandrov-Fenchel-type inequalities in the unit ball and half-space.