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

TL;DR

This paper establishes the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball by introducing quermassintegrals and employing a specialized curvature flow, extending previous free boundary results.

Contribution

It introduces quermassintegrals for hypersurfaces with capillary boundary and proves a new inequality using a nonlinear curvature flow, generalizing prior free boundary results.

Findings

Derived the first variational formula for quermassintegrals with capillary boundary.

Established the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.

Extended previous free boundary results to capillary boundary cases.

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 $\mathbb{B}^{n+1}$.