Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary II

TL;DR

This paper proves a conjecture related to Alexandrov-Fenchel inequalities for convex hypersurfaces with capillary boundary in the half-space, establishing a general theory for capillary convex bodies and their mixed volumes.

Contribution

It provides a proof of a conjecture on Alexandrov-Fenchel inequalities for capillary convex bodies in the half-space, extending the theory to mixed volumes.

Findings

Proved the conjecture on Alexandrov-Fenchel inequalities for capillary hypersurfaces.

Established a general theory for capillary convex bodies in the half-space.

Derived inequalities for mixed volumes of capillary convex bodies.

Abstract

In this paper, we provide an affirmative answer to [16, Conjecture 1.5] on the Alexandrov-Fenchel inequality for quermassintegrals for convex capillary hypersurfaces in the Euclidean half-space. More generally, we establish a theory for capillary convex bodies in the half-space and prove a general Alexandrov-Fenchel inequality for mixed volumes of capillary convex bodies. The conjecture [16, Conjecture 1.5] follows as its consequence.