Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball

Abstract

In this paper, we first introduce the quermassintegrals for convex hypersurfaces with capillary boundary in the unit Euclidean ball $\mathbb{B}^{n+1}$ and derive its first variational formula. Then by using a locally constrained nonlinear curvature flow, which preserves the $n$-th quermassintegral and non-decreases the $k$-th quermassintegral, we obtain the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in $\mathbb{B}^{n+1}$. This generalizes the result in \cite{SWX} for convex hypersurfaces with free boundary in $\mathbb{B}^{n+1}$.