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

Xinqun Mei, Guofang Wang, Liangjun Weng, Chao Xia

TL;DR

The paper resolves the Alexandrov-Fenchel-type inequalities for mixed volumes of capillary convex bodies in the Euclidean half-space with a fixed contact angle $\theta$, affirming Conjecture 1.5 from prior work. It develops a capillary convex-body framework via the capillary Gauss map to $\mathcal{C}_{\theta}$ and capillary support functions $h$, showing quermassintegrals arise as special mixed volumes and thereby reducing the AF inequality to the classical form. The main tool is a spectral approach: a self-adjoint operator $\mathcal{A}$ with Robin boundary on $\mathcal{C}_{\theta}$ yields a sharp lower bound that forces the positive eigenspace to be one-dimensional, giving rigidity and equality characterizations. Consequences include a full capillary AF inequality with rigidity (equality only for spherical caps), a Steiner-type expansion and Minkowski-type interpretations for capillary convex bodies, and a resolution of the conjecture for all $\theta\in(0,\pi)$.

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.

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

TL;DR

The paper resolves the Alexandrov-Fenchel-type inequalities for mixed volumes of capillary convex bodies in the Euclidean half-space with a fixed contact angle , affirming Conjecture 1.5 from prior work. It develops a capillary convex-body framework via the capillary Gauss map to and capillary support functions , showing quermassintegrals arise as special mixed volumes and thereby reducing the AF inequality to the classical form. The main tool is a spectral approach: a self-adjoint operator with Robin boundary on yields a sharp lower bound that forces the positive eigenspace to be one-dimensional, giving rigidity and equality characterizations. Consequences include a full capillary AF inequality with rigidity (equality only for spherical caps), a Steiner-type expansion and Minkowski-type interpretations for capillary convex bodies, and a resolution of the conjecture for all .

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.
Paper Structure (5 sections, 17 theorems, 90 equations)

This paper contains 5 sections, 17 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Capillary convex bodies and mixed volume
  3. Capillary convex bodies and capillary convex functions
  4. Mixed volume for capillary convex bodies
  5. The Alexandrov-Fenchel inequalities for mixed volume

Key Result

Theorem 1.1

Let $\widehat{\Sigma}_i\in\mathcal{K}_{\theta}$ for $1\leq i\leq n+1$. Then Equality holds if and only if the capillary support functions $h_j$ of $\widehat{\Sigma}_j, j=1, 2$, satisfy for some constants $a, a_{i} \in \mathbb{R}$, $i=1,\cdots, n$, and $\{E_i\}_{i=1}^n$ the horizontal coordinate unit vectors of $\overline{\mathbb{R}^{n+1}_+}$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6
  • ...and 25 more