Capillary Orlicz-Minkowski Inequality

• Capillary Orlicz-Minkowski inequality is the half-space analogue of the classical Orlicz-Minkowski inequality, linking capillary mixed volumes with enclosed Euclidean volumes.
• It employs a geometric framework in the upper half-space using capillary support functions and Robin boundary conditions to characterize strictly convex capillary hypersurfaces.
• The proof leverages Jensen’s and Alexandrov-Fenchel inequalities, establishing both sharp lower bounds and rigidity via spherical cap extremals.

The capillary Orlicz-Minkowski inequality is the half-space, capillary analogue of the classical Orlicz-Minkowski inequality in convex geometry. In its current formulation, it concerns strictly convex capillary hypersurfaces in the upper Euclidean half-space, encoded by a support function satisfying a Robin boundary condition, and it gives a lower bound for a capillary Orlicz mixed volume in terms of the enclosed Euclidean volumes. Its sharp equality case is dilation rigidity, while spherical caps serve as the canonical normalized extremals and the model solutions in the associated capillary Orlicz-Minkowski problem (Wang et al., 13 Sep 2025).

1. Geometric framework in the upper half-space

The ambient space is the upper Euclidean half-space

$$\mathbb{R}^{n+1}_+=\{x\in\mathbb{R}^{n+1}:x_{n+1}>0\},$$

with vertical vector $e=(0,\dots,0,-1)$ or $e:=-E_{n+1}$, depending on notation. A $C^2$-smooth, strictly convex hypersurface $\Sigma\subset \mathbb{R}^{n+1}_+$ with boundary on $\partial\mathbb{R}^{n+1}_+$ is capillary if it meets the boundary plane at a constant contact angle $\theta\in(0,\pi)$, expressed by

$\cos(\pi-\theta)=\langle \nu,e\rangle$

along $\partial\Sigma$, where $\nu$ is the outward unit normal. The body enclosed by $e=(0,\dots,0,-1)$0 is a capillary convex body; the class of such bodies is denoted $e=(0,\dots,0,-1)$1, and $e=(0,\dots,0,-1)$2 denotes the subclass whose flat boundary contains the origin in its interior (Wang et al., 13 Sep 2025).

The model geometry is provided by the spherical cap

$e=(0,\dots,0,-1)$3

with unit cap $e=(0,\dots,0,-1)$4. For a strictly convex capillary hypersurface, the capillary Gauss map

$e=(0,\dots,0,-1)$5

is a diffeomorphism. Using the inverse capillary Gauss map, one defines the support function

$e=(0,\dots,0,-1)$6

and the distinguished reference function

$e=(0,\dots,0,-1)$7

which is the support function of the spherical cap $e=(0,\dots,0,-1)$8. The capillary support function is then

$e=(0,\dots,0,-1)$9

A defining structural feature of the capillary theory is the Robin boundary condition

$e:=-E_{n+1}$0

where $e:=-E_{n+1}$1 is the outward unit co-normal of $e:=-E_{n+1}$2. This boundary condition replaces the boundary-free support-function formalism used for closed convex bodies. The same framework also supports an evenness notion: a function on $e:=-E_{n+1}$3 is even if $e:=-E_{n+1}$4, where $e:=-E_{n+1}$5, and a capillary body is symmetric if its support function is even (Wang et al., 13 Sep 2025).

2. Orlicz data, capillary measures, and mixed volume

The Orlicz input is taken from a class $e:=-E_{n+1}$6 of $e:=-E_{n+1}$7, strictly increasing, convex, log-concave functions

$e:=-E_{n+1}$8

subject to

$e:=-E_{n+1}$9

$C^2$0

$C^2$1

A model example is $C^2$2 with $C^2$3 (Wang et al., 13 Sep 2025).

For $C^2$4, the capillary surface area measure is

$C^2$5

and its total mass is the wetting energy

$C^2$6

The capillary Orlicz surface area measure is defined by

$C^2$7

When $C^2$8, this reduces to the capillary $C^2$9 surface area measure

$\Sigma\subset \mathbb{R}^{n+1}_+$0

The capillary Orlicz mixed volume of $\Sigma\subset \mathbb{R}^{n+1}_+$1 is

$\Sigma\subset \mathbb{R}^{n+1}_+$2

and the associated capillary Orlicz volume is $\Sigma\subset \mathbb{R}^{n+1}_+$3. A related cone-volume measure is

$\Sigma\subset \mathbb{R}^{n+1}_+$4

These definitions supply the exact measure-theoretic objects appearing in the capillary Orlicz-Minkowski and capillary Orlicz-Brunn-Minkowski theories (Wang et al., 13 Sep 2025).

3. Statement of the capillary Orlicz-Minkowski inequality

For $\Sigma\subset \mathbb{R}^{n+1}_+$5 and $\Sigma\subset \mathbb{R}^{n+1}_+$6, the capillary Orlicz-Minkowski inequality is

$\Sigma\subset \mathbb{R}^{n+1}_+$7

Equality holds if $\Sigma\subset \mathbb{R}^{n+1}_+$8 and $\Sigma\subset \mathbb{R}^{n+1}_+$9 are dilates; if $\partial\mathbb{R}^{n+1}_+$0 is strictly convex, then equality holds if and only if they are dilates (Wang et al., 13 Sep 2025).

This is the precise capillary analogue of the Orlicz-Minkowski inequality. The volume term is the Euclidean volume of the capillary convex body, while the mixed-volume term is computed on the capillary spherical domain $\partial\mathbb{R}^{n+1}_+$1 through the support function and the Monge-Ampère-type determinant $\partial\mathbb{R}^{n+1}_+$2. In the normalized capillary setting, the spherical cap is the reference body: for $\partial\mathbb{R}^{n+1}_+$3, one has $\partial\mathbb{R}^{n+1}_+$4, and the formulas reduce to the equality model (Wang et al., 13 Sep 2025).

The $\partial\mathbb{R}^{n+1}_+$5 theory is obtained by the specialization $\partial\mathbb{R}^{n+1}_+$6. In that case, the capillary Orlicz combination becomes the capillary $\partial\mathbb{R}^{n+1}_+$7-combination, the capillary Orlicz mixed volume becomes the capillary $\partial\mathbb{R}^{n+1}_+$8 mixed volume, and the capillary Orlicz-Minkowski inequality reduces to the capillary $\partial\mathbb{R}^{n+1}_+$9-Minkowski inequality. A later flow treatment establishes smooth solutions to the capillary even $\theta\in(0,\pi)$0 Minkowski problem for all $\theta\in(0,\pi)$1 and to the capillary $\theta\in(0,\pi)$2 Minkowski problem for $\theta\in(0,\pi)$3, again in the upper half-space setting (Hu et al., 7 Sep 2025).

4. Pre-Orlicz and anisotropic antecedents

The immediate geometric precursor is a Minkowski-type inequality for star-shaped and mean convex capillary hypersurfaces in a half-space. For a capillary hypersurface $\theta\in(0,\pi)$4, the capillary area functional and the second capillary quermassintegral are

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

$\theta\in(0,\pi)$6

The sharp inequality is

$\theta\in(0,\pi)$7

with equality if and only if $\theta\in(0,\pi)$8 is a capillary spherical cap. Although this result is not formulated in Orlicz language, it was explicitly identified as exactly of the type one would want as a starting point for a capillary Orlicz-Minkowski theory, because it relates a curvature integral to a boundary-corrected area functional and has spherical-cap rigidity (Wang et al., 2022).

A second antecedent is anisotropic. For anisotropic $\theta\in(0,\pi)$9-capillary hypersurfaces in the half-space, a generalized Hsiung-Minkowski integral formula yields a weighted capillary Orlicz-Minkowski-type inequality. With anisotropic capillary support function

$\cos(\pi-\theta)=\langle \nu,e\rangle$0

Corollary 1.5 states that for any smooth $\cos(\pi-\theta)=\langle \nu,e\rangle$1,

$\cos(\pi-\theta)=\langle \nu,e\rangle$2

if $\cos(\pi-\theta)=\langle \nu,e\rangle$3, while the inequality is reversed if $\cos(\pi-\theta)=\langle \nu,e\rangle$4. Equality holds if and only if either $\cos(\pi-\theta)=\langle \nu,e\rangle$5 is constant or $\cos(\pi-\theta)=\langle \nu,e\rangle$6 is an $\cos(\pi-\theta)=\langle \nu,e\rangle$7-capillary Wulff shape. This is the mechanism used there to obtain uniqueness for the anisotropic Orlicz-Christoffel-Minkowski problem and a new proof of uniqueness for the $\cos(\pi-\theta)=\langle \nu,e\rangle$8-Minkowski problem with $\cos(\pi-\theta)=\langle \nu,e\rangle$9 in Euclidean capillary convex bodies geometry (Gao et al., 2024).

These two lines of development show that capillary Minkowski theory has both isotropic and anisotropic branches. A plausible implication is that the current capillary Orlicz-Minkowski inequality sits at the intersection of a convex-body support-function formalism and an older integral-identity tradition.

5. Proof architecture, variational identities, and Brunn-Minkowski equivalence

The proof of the capillary Orlicz-Minkowski inequality is a two-step argument. First, Jensen’s inequality is applied to the cone-volume measure $\partial\Sigma$0, which yields

$\partial\Sigma$1

Second, the capillary Alexandrov-Fenchel inequality implies the capillary Minkowski inequality

$\partial\Sigma$2

Since $\partial\Sigma$3 is increasing, the stated Orlicz inequality follows. The equality case is inherited from equality in Jensen’s inequality together with equality in the Alexandrov-Fenchel inequality (Wang et al., 13 Sep 2025).

The same work defines the capillary Orlicz combination

$\partial\Sigma$4

and proves the capillary Orlicz-Brunn-Minkowski inequality

$\partial\Sigma$5

The paper further shows that the capillary Orlicz-Minkowski and capillary Orlicz-Brunn-Minkowski inequalities are equivalent via a standard perturbation/variational argument (Wang et al., 13 Sep 2025).

The variational structure is explicit. The derivative of the volume along Orlicz perturbations is

$\partial\Sigma$6

This identity is the basis for defining the Orlicz mixed volume and proving the inequality (Wang et al., 13 Sep 2025).

A related but non-capillary prototype is the $\partial\Sigma$7-capacitary Orlicz-Minkowski theory. There, the Orlicz $\partial\Sigma$8 mixed $\partial\Sigma$9-capacity

$\nu$0

satisfies

$\nu$1

again proved by Jensen plus a Minkowski inequality, and again equivalent in an appropriate sense to an Orlicz-Brunn-Minkowski inequality (Hong et al., 2017). This provides a direct external analogue for the capillary theory.

6. Existence theory, flow methods, and current scope

The inequality theory in the capillary setting is tightly connected to existence and rigidity for prescribed curvature problems. For the capillary even Orlicz-Minkowski problem, if $\nu$2, $\nu$3, and $\nu$4 is positive and even and satisfies

$\nu$5

then there exists a smooth symmetric $\nu$6 with $\nu$7 solving

$\nu$8

If equality holds in the integral condition and $\nu$9 is strictly convex, then

$e=(0,\dots,0,-1)$00

so the capillary convex body must be a spherical cap. This is the main rigidity consequence of the capillary Orlicz-Minkowski inequality in that setting (Wang et al., 13 Sep 2025).

A subsequent flow formulation studies the more general capillary Orlicz-Minkowski problem

$e=(0,\dots,0,-1)$01

and uses the anisotropic capillary Gauss curvature flow

$e=(0,\dots,0,-1)$02

with the same Robin boundary condition. Under the growth condition

$e=(0,\dots,0,-1)$03

the flow has a smooth, strictly convex solution for all $e=(0,\dots,0,-1)$04, and some subsequence converges in $e=(0,\dots,0,-1)$05 to a smooth strictly convex capillary hypersurface solving the capillary Orlicz-Minkowski equation. The associated Lyapunov functional is

$e=(0,\dots,0,-1)$06

and it satisfies

$e=(0,\dots,0,-1)$07

That work explicitly states that it is primarily a flow/existence paper and does not appear to prove a new capillary Orlicz-Minkowski inequality in the sense of a sharp geometric inequality with equality characterization (Li et al., 21 Jan 2026).

For the $e=(0,\dots,0,-1)$08 specialization, the capillary $e=(0,\dots,0,-1)$09 Minkowski flow

$e=(0,\dots,0,-1)$10

produces smooth solutions to

$e=(0,\dots,0,-1)$11

for $e=(0,\dots,0,-1)$12 without evenness, and to the even normalized problem for all $e=(0,\dots,0,-1)$13 under evenness assumptions. The same source notes that for the regular capillary Minkowski problem, existence and uniqueness for $e=(0,\dots,0,-1)$14 had been obtained earlier by the continuity method, while the case $e=(0,\dots,0,-1)$15 remains open (Hu et al., 7 Sep 2025).

Taken together, these results identify the capillary Orlicz-Minkowski inequality as part of a broader half-space theory in which Robin boundary geometry, spherical-cap normalization, mixed-volume inequalities, and curvature flows all interact. The established convex theory is sharp and rigid; the nonconvex star-shaped theory is presently available at the Minkowski-type level; and the flow literature indicates that existence theory can extend beyond the even setting even when the corresponding sharp inequality is not yet formulated in full generality (Wang et al., 2022).