Capillary Orlicz-Brunn-Minkowski Inequality

• The topic defines capillary convex bodies in the upper half-space with Robin boundary conditions and establishes an Orlicz-type volume inequality.
• It employs an Orlicz combination of support functions on a spherical cap to relate volumes under strict convexity assumptions.
• The approach integrates variational techniques like Aleksandrov-Fenchel and Jensen’s inequalities to demonstrate rigidity and existence results.

The capillary Orlicz-Brunn-Minkowski inequality is a Brunn-Minkowski-type inequality for capillary convex bodies in the upper Euclidean half-space, formulated in terms of an Orlicz combination of support functions on a spherical cap and adapted to a constant-contact-angle, or Robin, boundary geometry. In the recent capillary Orlicz-Minkowski theory, it appears as the volume comparison principle associated with a Robin boundary analogue of the Orlicz-Minkowski problem: given two capillary convex bodies and an admissible Orlicz function, the volume of their capillary Orlicz combination controls the volumes of the original bodies through a sharp inequality, with equality characterized by dilations under strict convexity assumptions (Wang et al., 13 Sep 2025). Conceptually, it is the capillary counterpart of the full-space Orlicz-Brunn-Minkowski inequality of Gardner, Hug, and Weil, but the relevant support functions are defined on a capillary spherical cap and satisfy a nonlinear boundary condition reflecting the prescribed contact angle (Gardner et al., 2013).

1. Geometric setting and admissible functions

The capillary framework is formulated in the closed upper Euclidean half-space $\overline{\mathbb{R}^{n+1}_+}$. A capillary convex body is a compact convex domain bounded by a smooth strictly convex hypersurface $\Sigma$ together with the flat boundary $x_{n+1}=0$, such that $\Sigma$ meets the hyperplane at a constant contact angle $\theta\in(0,\pi)$. Its support function is defined on a spherical cap $\mathcal{C}_\theta$, the model capillary convex surface. The notation $\mathcal{K}_\theta^\circ$ denotes capillary convex bodies whose flat boundary contains the origin in its interior (Wang et al., 13 Sep 2025).

The Orlicz data are encoded by a function $\phi$ in the class $\mathcal{O}$, consisting of $C^2$-smooth, strictly increasing, convex, log-concave functions $\Sigma$0 with $\Sigma$1, together with additional growth and monotonicity conditions. The summary states in particular that

$\Sigma$2

and that $\Sigma$3 with $\Sigma$4 is admissible (Wang et al., 13 Sep 2025).

A defining geometric feature of the capillary setting is the Robin boundary condition for the support function,

$\Sigma$5

where $\Sigma$6 is the co-normal along the boundary of the spherical cap. This boundary operator replaces the unrestricted Euclidean support-function formalism of the full-space Orlicz theory and is responsible for much of the analytic complexity of the capillary case (Wang et al., 13 Sep 2025).

2. Capillary Orlicz combination and the main inequality

Given $\Sigma$7, coefficients $\Sigma$8 not both zero, and capillary convex bodies $\Sigma$9, the capillary Orlicz combination $x_{n+1}=0$0 is defined by its support function: $x_{n+1}=0$1 This is the capillary analogue of Orlicz addition via support functions, now constrained to the spherical-cap domain (Wang et al., 13 Sep 2025).

The capillary Orlicz-Brunn-Minkowski inequality, stated as Theorem 3.4 in the cited work, is

$x_{n+1}=0$2

where $x_{n+1}=0$3 and $x_{n+1}=0$4 denotes volume (Wang et al., 13 Sep 2025).

For strictly convex $x_{n+1}=0$5, equality holds if and only if $x_{n+1}=0$6 and $x_{n+1}=0$7 are dilates of each other. This is the standard rigidity pattern of Orlicz-Brunn-Minkowski theory, but in the capillary setting the notion of dilation is understood within the upper-half-space capillary geometry (Wang et al., 13 Sep 2025).

Geometrically, the inequality relates the volumes of capillary convex bodies and their Orlicz combination in direct analogy with the way the classical Brunn-Minkowski inequality relates the volumes of convex bodies and their Minkowski sums. The novelty lies in the fact that the combination is not defined on the full sphere and is compatible with the capillary boundary condition rather than with unconstrained Euclidean addition (Wang et al., 13 Sep 2025).

3. Associated capillary Orlicz-Minkowski inequality and proof structure

The capillary Orlicz-Brunn-Minkowski inequality is derived from a capillary Orlicz-Minkowski inequality. The latter is stated in the same paper as

$x_{n+1}=0$8

with

$x_{n+1}=0$9

This functional is the capillary Orlicz mixed volume, expressed through the support function on $\Sigma$0 and the capillary Monge-Ampère density (Wang et al., 13 Sep 2025).

The proof architecture described for the capillary theory has three layers. First, the capillary Orlicz-Minkowski inequality is established by combining the Aleksandrov-Fenchel inequality for capillary convex bodies with Jensen’s inequality. Second, the capillary Orlicz-Brunn-Minkowski inequality is deduced by applying variational arguments to Orlicz combinations and first variations of volume with respect to perturbations of the support function. Third, the boundary condition is handled by a dedicated capillary computation: Lemma 3.2 verifies that the capillary Orlicz combination remains compatible with the Robin boundary condition, using structural properties of $\Sigma$1, notably the condition denoted $\Sigma$2 in the paper (Wang et al., 13 Sep 2025).

This proof strategy parallels the standard Orlicz-Brunn-Minkowski mechanism—mixed functional first, Brunn-Minkowski consequence second—but the capillary version must accommodate the restricted domain $\Sigma$3, the Robin boundary operator, and boundary terms inside the capillary Aleksandrov-Fenchel theory. A common misconception is that the capillary inequality is obtained by a formal substitution of a spherical cap for the full sphere; the capillary proof instead depends on boundary PDE estimates specific to the Robin setting (Wang et al., 13 Sep 2025).

4. Relation to the capillary Orlicz-Minkowski problem and flow methods

The inequality is embedded in a broader existence theory for the capillary Orlicz-Minkowski problem. In one formulation, for a smooth positive function $\Sigma$4 on $\Sigma$5 and a function $\Sigma$6, the problem is to find a strictly convex capillary hypersurface with support function $\Sigma$7 satisfying

$\Sigma$8

where $\Sigma$9 (Li et al., 21 Jan 2026).

The continuity-method paper on the capillary Orlicz-Minkowski problem introduces the Robin boundary analogue of the Orlicz-Minkowski problem and obtains volume-normalized smooth solutions to the capillary even Orlicz-Minkowski problem. Within that framework it also establishes the capillary Orlicz-Brunn-Minkowski inequality and the capillary Orlicz-Minkowski inequality (Wang et al., 13 Sep 2025).

A later flow-based approach studies the anisotropic capillary Gauss curvature flow

$\theta\in(0,\pi)$0

and proves long-time existence and convergence to a stationary solution under natural structural assumptions. The stationary points of the flow are precisely solutions of the capillary Orlicz-Minkowski equation, yielding smooth strictly convex solutions without the evenness assumption (Li et al., 21 Jan 2026).

The central Lyapunov functional is

$\theta\in(0,\pi)$1

with dissipation identity

$\theta\in(0,\pi)$2

The summary explicitly notes that the paper does not state a new Orlicz-Brunn-Minkowski inequality, but that the monotonicity formula in Lemma 4.1 can be interpreted as providing a generalized capillary Orlicz-Brunn-Minkowski functional (Li et al., 21 Jan 2026). This suggests a variational perspective in which capillary Brunn-Minkowski-type inequalities arise as optimality statements for $\theta\in(0,\pi)$3 over capillary convex bodies.

5. Position within Orlicz-Brunn-Minkowski theory

The capillary inequality is best understood as a boundary-value extension of the general Orlicz-Brunn-Minkowski program. In the full-space theory, Orlicz addition of convex sets is defined by

$\theta\in(0,\pi)$4

and the associated Brunn-Minkowski inequality is

$\theta\in(0,\pi)$5

with equality for strictly convex $\theta\in(0,\pi)$6 precisely when $\theta\in(0,\pi)$7 and $\theta\in(0,\pi)$8 are dilates (Gardner et al., 2013).

The capillary formula replaces the full-sphere support-function relation by a spherical-cap support-function relation and replaces unconstrained Euclidean geometry by a capillary geometry carrying a Robin boundary condition. The capillary theory also recovers the capillary $\theta\in(0,\pi)$9-Brunn-Minkowski inequality when $\mathcal{C}_\theta$0, and if the capillary angle is sent to $\mathcal{C}_\theta$1, one recovers the full-space Orlicz-Brunn-Minkowski theory (Wang et al., 13 Sep 2025).

Within the broader literature, the same Orlicz paradigm has been extended to several other geometric functionals: affine quermassintegrals (Dafnis, 2018), dual mixed volumes and Orlicz harmonic addition (Zhao, 2020), dual Orlicz $\mathcal{C}_\theta$2 affine and geominimal surface areas for star bodies (Ye, 2014), general volumes and dual Orlicz curvature measures (Gardner et al., 2018), and mixed width integrals (Zhao, 2021). The capillary inequality belongs to this family but is distinguished by the capillary boundary geometry and by its direct interaction with boundary Monge-Ampère equations and capillary curvature flows.

6. Equality, normalization, and present scope

The sharp equality condition in the capillary Orlicz-Brunn-Minkowski inequality is the same structural rigidity that appears throughout Orlicz-Brunn-Minkowski theory: if $\mathcal{C}_\theta$3 is strictly convex, equality holds if and only if the two capillary convex bodies are dilates (Wang et al., 13 Sep 2025). In this sense, spherical-cap geometry does not alter the extremal class, but it does alter the analytic mechanism used to reach rigidity.

Normalization plays a central role. The continuity-method paper seeks volume-normalized smooth solutions of the capillary even Orlicz-Minkowski problem, while the flow paper produces smooth strictly convex solutions without the evenness assumption and establishes subsequential $\mathcal{C}_\theta$4 convergence of the evolving support functions to stationary solutions (Wang et al., 13 Sep 2025, Li et al., 21 Jan 2026). A plausible implication is that the capillary Orlicz-Brunn-Minkowski inequality should be read not only as an isolated volume inequality but also as part of a normalization and compactness framework for the underlying capillary Minkowski problem.

The current scope of the theory is therefore twofold. On the geometric side, it supplies a sharp Orlicz-type volume inequality for capillary convex bodies in the upper half-space. On the analytic side, it is intertwined with Robin-Monge-Ampère equations, continuity arguments, and anisotropic capillary Gauss curvature flows. The available results establish the inequality itself, its equality case, the even-data continuity-method existence theory, and a non-even flow-based existence theory; they also indicate that monotonicity formulas in the flow setting encode a broader capillary Orlicz-Brunn-Minkowski variational structure (Wang et al., 13 Sep 2025, Li et al., 21 Jan 2026).