Alexandrov-Fenchel inequalities for convex anisotropic capillary hypersurfaces in the half-space

TL;DR

This paper extends Alexandrov-Fenchel inequalities to anisotropic capillary convex bodies in the half-space, broadening the scope of previous results and weakening existing conditions in the inequalities.

Contribution

It develops a new theory for anisotropic capillary convex bodies and establishes a general Alexandrov-Fenchel inequality in this setting.

Findings

Extended inequalities to anisotropic setting

Weakened conditions compared to previous results

Established a general inequality for mixed volumes

Abstract

In this paper, the results of Mei, Wang, Weng and Xia [Math. Z., 2025, MR4911815] on capillary convex bodies are extended to the anisotropic setting. We develop a theory for anisotropic capillary convex bodies in the half-space and establish a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies. Thus, this weakens the conditions of the inequality in [Ding-Gao-Li,arXiv:2408.10740, Theorem 1.4] and extends it to a more general case.