Anisotropic Capillary Convex Bodies

• Anisotropic capillary convex bodies are convex shapes defined by direction-dependent surface energies that yield equilibrium configurations like full-space Wulff shapes and truncated versions in constrained geometries.
• They are characterized through advanced tools such as the Cahn–Hoffman map, anisotropic curvature measures, and Minkowski-type formulas, linking geometric invariants with capillary conditions.
• Applications include the study of anisotropic curvature flows, rigidity theorems, and capillary boundary problems in half-spaces, providing a comprehensive framework for anisotropic convex geometry.

Anisotropic capillary convex bodies are convex bodies whose interfacial energy is governed by a direction-dependent surface tension rather than a constant one, so that the distinguished equilibrium shapes are Wulff shapes in the full space and truncated Wulff shapes in a half-space or other constrained geometries. In the smooth setting, the subject is formulated through the Cahn–Hoffman map, anisotropic principal curvatures, and higher-order anisotropic mean curvatures; in the nonsmooth convex setting, it is encoded by anisotropic support, area, and curvature measures arising from anisotropic Steiner formulas and mixed volumes (Santilli, 2021, Gao et al., 2024).

1. Anisotropic surface energy and Wulff geometry

Let $C\subset \mathbb{R}^{n+1}$ be a convex body. In one standard formulation, the anisotropy is a uniformly convex $C^2$-norm $\varphi$ on $\mathbb{R}^{n+1}$, with dual norm

$\varphi^*(u)=\sup\{v\cdot u : \varphi(v)=1\}.$

The associated Wulff shape is

$$W_\varphi=\partial B_\varphi^*,\qquad B_\varphi^*=\{x\in\mathbb{R}^{n+1}:\varphi^*(x)\le 1\}.$$

If $\varphi$ is uniformly convex and $C^2$, then $W_\varphi$ is a smooth, strictly convex hypersurface, and its Euclidean unit normal is a $C^1$-diffeomorphism onto $C^2$0. The corresponding Cahn–Hoffman map $C^2$1 satisfies

$C^2$2

This realizes the geometric duality between the anisotropy and its Wulff crystal (Santilli, 2021).

For a convex body $C^2$3, the anisotropic perimeter, or anisotropic surface energy, is

$C^2$4

and when $C^2$5 comes from a support function $C^2$6 on $C^2$7, this is equivalent to

$C^2$8

The minimizer of $C^2$9 under a volume constraint is, up to translation and scaling, the Wulff shape $\varphi$0. In capillarity language, Wulff shapes model equilibrium shapes of anisotropic liquid droplets or crystals (Santilli, 2021).

A parallel convex-analytic formulation starts from a convex body $\varphi$1 with $\varphi$2. Its gauge $\varphi$3, support function $\varphi$4, and polar body $\varphi$5 satisfy

$\varphi$6

For a set of finite perimeter $\varphi$7, the anisotropic surface energy associated with $\varphi$8 is

$\varphi$9

and the anisotropic isoperimetric inequality states

$\mathbb{R}^{n+1}$0

with equality if and only if $\mathbb{R}^{n+1}$1 is, up to translation and null sets, a dilate of $\mathbb{R}^{n+1}$2. This is the Wulff inequality in convex-geometric form (Bianchi et al., 2024).

In the half-space setting, an anisotropic capillary convex body is modeled by a bounded domain $\mathbb{R}^{n+1}$3 whose free boundary $\mathbb{R}^{n+1}$4 is a strictly convex anisotropic capillary hypersurface and whose remaining boundary lies in the supporting hyperplane. The canonical model is a truncated Wulff shape

$\mathbb{R}^{n+1}$5

with $\mathbb{R}^{n+1}$6 chosen so that the anisotropic capillary boundary condition is satisfied (Gao et al., 2024).

2. Curvature, support functions, and measure-theoretic invariants

For a smooth convex hypersurface, anisotropic curvature is defined through the anisotropic normal. With anisotropy $\mathbb{R}^{n+1}$7 satisfying

$\mathbb{R}^{n+1}$8

the Cahn–Hoffman map is

$\mathbb{R}^{n+1}$9

the anisotropic Gauss map is

$\varphi^*(u)=\sup\{v\cdot u : \varphi(v)=1\}.$0

and the anisotropic Weingarten map is

$\varphi^*(u)=\sup\{v\cdot u : \varphi(v)=1\}.$1

Its eigenvalues $\varphi^*(u)=\sup\{v\cdot u : \varphi(v)=1\}.$2 are the anisotropic principal curvatures, and the normalized anisotropic $\varphi^*(u)=\sup\{v\cdot u : \varphi(v)=1\}.$3-th mean curvatures are

$\varphi^*(u)=\sup\{v\cdot u : \varphi(v)=1\}.$4

In the norm language of $\varphi^*(u)=\sup\{v\cdot u : \varphi(v)=1\}.$5, the anisotropic mean curvatures $\varphi^*(u)=\sup\{v\cdot u : \varphi(v)=1\}.$6 are the symmetric functions of the anisotropic principal curvatures $\varphi^*(u)=\sup\{v\cdot u : \varphi(v)=1\}.$7 (Gao et al., 2024, Santilli, 2021).

Beyond smooth boundaries, several measure-theoretic frameworks have been developed. One construction starts from the anisotropic distance

$\varphi^*(u)=\sup\{v\cdot u : \varphi(v)=1\}.$8

the anisotropic normal bundle

$\varphi^*(u)=\sup\{v\cdot u : \varphi(v)=1\}.$9

and generalized anisotropic principal curvatures on $$W_\varphi=\partial B_\varphi^*,\qquad B_\varphi^*=\{x\in\mathbb{R}^{n+1}:\varphi^*(x)\le 1\}.$$0. For convex bodies, this yields anisotropic curvature measures $$W_\varphi=\partial B_\varphi^*,\qquad B_\varphi^*=\{x\in\mathbb{R}^{n+1}:\varphi^*(x)\le 1\}.$$1 and a local Steiner formula

$$W_\varphi=\partial B_\varphi^*,\qquad B_\varphi^*=\{x\in\mathbb{R}^{n+1}:\varphi^*(x)\le 1\}.$$2

for Borel $$W_\varphi=\partial B_\varphi^*,\qquad B_\varphi^*=\{x\in\mathbb{R}^{n+1}:\varphi^*(x)\le 1\}.$$3. In the smooth case,

$$W_\varphi=\partial B_\varphi^*,\qquad B_\varphi^*=\{x\in\mathbb{R}^{n+1}:\varphi^*(x)\le 1\}.$$4

and, in particular,

$$W_\varphi=\partial B_\varphi^*,\qquad B_\varphi^*=\{x\in\mathbb{R}^{n+1}:\varphi^*(x)\le 1\}.$$5

so $$W_\varphi=\partial B_\varphi^*,\qquad B_\varphi^*=\{x\in\mathbb{R}^{n+1}:\varphi^*(x)\le 1\}.$$6 is the anisotropic surface energy measure (Santilli, 2021).

A second construction, due to Huang, Li, Xiao, and Zhou, defines anisotropic curvature measures $$W_\varphi=\partial B_\varphi^*,\qquad B_\varphi^*=\{x\in\mathbb{R}^{n+1}:\varphi^*(x)\le 1\}.$$7 as the coefficients in the local anisotropic Steiner formula

$$W_\varphi=\partial B_\varphi^*,\qquad B_\varphi^*=\{x\in\mathbb{R}^{n+1}:\varphi^*(x)\le 1\}.$$8

where $$W_\varphi=\partial B_\varphi^*,\qquad B_\varphi^*=\{x\in\mathbb{R}^{n+1}:\varphi^*(x)\le 1\}.$$9 is a local anisotropic $\varphi$0-parallel set. The total measures satisfy

$\varphi$1

the anisotropic mixed volumes relative to the Wulff shape $\varphi$2, and

$\varphi$3

is exactly the anisotropic surface energy (Andrews et al., 2021).

A third, relative-differential-geometric framework fixes a gauge body $\varphi$4, regular and strictly convex, and defines anisotropic area measures on $\varphi$5 by

$\varphi$6

together with anisotropic support measures $\varphi$7 and anisotropic curvature measures

$\varphi$8

In the $\varphi$9 setting, these are represented by the symmetric functions $C^2$0 of the relative principal radii: $C^2$1 This framework is explicitly tied to relative normalization and gauge-body geometry (Schneider, 10 Jun 2025).

The coexistence of $C^2$2, $C^2$3, and $C^2$4 is not a contradiction. It reflects different normalizations and geometric models—norm-based anisotropy, Wulff-relative mixed volumes, and relative differential geometry—all organized around anisotropic parallel sets and mixed-volume expansions. This suggests that anisotropic capillary convex bodies are naturally studied through several complementary measure theories rather than through a single canonical formalism.

3. Rigidity in the full space: Wulff shapes, tangential bodies, and curvature relations

A central rigidity theorem states that a convex body whose anisotropic curvature measure is proportional to anisotropic perimeter must be a Wulff shape. More precisely, if $C^2$5 is a convex body, $C^2$6 is a uniformly convex $C^2$7-norm, $C^2$8, and

$C^2$9

as Radon measures on $W_\varphi$0, then there exist $W_\varphi$1 and $W_\varphi$2 such that

$W_\varphi$3

Equivalently, for any $W_\varphi$4, if

$W_\varphi$5

then $W_\varphi$6 is a translate and homothety of the Wulff shape. This theorem generalizes Schneider’s Euclidean characterization of balls and resolves the conjecture of Andrews and Wei (Santilli, 2021).

The proof combines an anisotropic Heintze–Karcher type inequality, anisotropic Minkowski formulas, and Newton–Maclaurin inequalities. Under the proportionality hypothesis one derives a lower bound for the first anisotropic mean curvature, then equality in the chain of inequalities forces anisotropic umbilicity,

$W_\varphi$7

and hence all anisotropic distance level sets are translates and homotheties of the Wulff shape (Santilli, 2021).

A related rigidity result uses the curvature measures $W_\varphi$8: if a convex body $W_\varphi$9 satisfies

$C^1$0

with $C^1$1, then $C^1$2 is a rescaled Wulff shape. In particular, $C^1$3 forces Wulff geometry. This extends classical results of Schneider and Kohlmann to the anisotropic setting and is used crucially in the long-time analysis of anisotropic volume-preserving curvature flows (Andrews et al., 2021).

By contrast, Schneider’s theory of anisotropic area measures yields a different rigidity class. If $C^1$4 is the gauge body and $C^1$5, then

$C^1$6

holds if and only if $C^1$7 is homothetic to a $C^1$8-tangential body of $C^1$9. Here the conclusion is not necessarily a Wulff shape but a tangential body determined by the gauge geometry (Schneider, 10 Jun 2025).

This distinction rules out a common oversimplification. Not every proportionality relation among anisotropic measures characterizes a Wulff shape. Proportionality between anisotropic curvature measures and anisotropic perimeter yields Wulff rigidity (Santilli, 2021), whereas proportionality between anisotropic area measures of orders $C^2$00 and $C^2$01 characterizes $C^2$02-tangential bodies of the gauge body (Schneider, 10 Jun 2025).

4. Half-space capillarity, truncated Wulff shapes, and rigidity

In the half-space $C^2$03, anisotropic capillarity is formulated for a compact orientable hypersurface $C^2$04 with $C^2$05, enclosing a bounded domain $C^2$06. If $C^2$07 is positive and $C^2$08, the anisotropic capillary boundary condition is

$C^2$09

or equivalently

$C^2$10

depending on notation. Geometrically, this prescribes a constant anisotropic contact angle along the boundary plane (Gao et al., 2024, Jia et al., 2022).

The model solutions are truncated Wulff shapes,

$C^2$11

also called $C^2$12-capillary Wulff shapes. Their anisotropic normal satisfies

$C^2$13

and the capillary condition fixes the vertical placement of the center $C^2$14 (Gao et al., 2024).

For anisotropic capillary hypersurfaces, a capillary support function is defined by

$C^2$15

A basic characterization asserts that $C^2$16 is a nonzero constant if and only if $C^2$17 is an $C^2$18-capillary Wulff shape. This makes $C^2$19 the anisotropic capillary analogue of the support function used in classical spherical-cap rigidity (Gao et al., 2024).

The integral-geometric core of the half-space theory is a generalized anisotropic Hsiung–Minkowski formula. For a smooth function $C^2$20 on $C^2$21 and $C^2$22,

$C^2$23

where $C^2$24 is the Newton transformation and $C^2$25 is an explicit tangential vector field. Setting $C^2$26 constant recovers the capillary Minkowski identity

$C^2$27

Together with the anisotropic Heintze–Karcher inequality

$C^2$28

with equality if and only if $C^2$29 is an $C^2$30-capillary Wulff shape, these identities drive a broad family of Alexandrov-type rigidity results (Gao et al., 2024).

One consequence is the anisotropic Alexandrov theorem in the half-space: any embedded anisotropic capillary hypersurface with constant anisotropic mean curvature is a truncated Wulff shape. The same conclusion holds for constant higher anisotropic mean curvature $C^2$31, $C^2$32, under the same capillarity assumptions (Jia et al., 2022).

A stronger classification is available at the level of stability. Any compact immersed anisotropic capillary constant anisotropic mean curvature hypersurface in the half-space is weakly stable if and only if it is a truncated Wulff shape. In dimension three, a stable anisotropic capillary minimal surface with Euclidean area growth is a half-plane (Guo et al., 2023).

The half-space theory also admits Willmore-type inequalities. For an unbounded closed convex set $C^2$33 and a compact embedded hypersurface $C^2$34 with $C^2$35 satisfying

$C^2$36

one has

$C^2$37

In a half-space, this yields the capillary inequality

$C^2$38

with equality if and only if $C^2$39 is an anisotropic $C^2$40-capillary Wulff cap (Jia et al., 2024).

5. Mixed volumes, capillary Minkowski problems, and support-function PDEs

Anisotropic capillary convex bodies are closely tied to support-function formulations of Minkowski-type problems. In the half-space, the capillary support-function framework is especially transparent in the Euclidean capillary theory. For a convex capillary hypersurface $C^2$41 with contact angle $C^2$42, the capillary Gauss map takes values in a spherical cap $C^2$43, and the support function $C^2$44 on $C^2$45 satisfies

$C^2$46

The capillary mixed volumes are then expressed by

$C^2$47

where $C^2$48 is the mixed discriminant. This leads to a theory of capillary quermassintegrals and a capillary Alexandrov–Fenchel inequality for mixed volumes of capillary convex bodies in the half-space (Mei et al., 2024).

That theory is isotropic, but it supplies an analytic template for the anisotropic case. This suggests that anisotropic capillary mixed volumes should be built from anisotropic support functions on an anisotropic cap domain, an anisotropic curvature operator replacing $C^2$49, and a Robin boundary condition expressing anisotropic Young’s law. The isotropic paper itself formulates this as a blueprint for extension rather than as an established anisotropic theorem (Mei et al., 2024).

In the fully anisotropic half-space theory, generalized Minkowski formulas already lead to uniqueness results for capillary curvature prescription problems. If $C^2$50 is strictly convex and anisotropic $C^2$51-capillary, and if

$C^2$52

for a smooth function $C^2$53 with $C^2$54, then $C^2$55 is an $C^2$56-capillary Wulff shape. This is the uniqueness statement for the anisotropic Orlicz–Christoffel–Minkowski problem in the capillary setting. In particular, when $C^2$57 with $C^2$58, it yields the corresponding uniqueness result for the capillary $C^2$59-Minkowski problem in Euclidean capillary convex bodies geometry (Gao et al., 2024).

More generally, linear and nonlinear curvature identities of the forms

$C^2$60

or

$C^2$61

with the monotonicity hypotheses stated in the source, force $C^2$62 to be an $C^2$63-capillary Wulff shape (Gao et al., 2024).

The capillary support-function PDE perspective also appears in newer work on capillary Minkowski problems in the half-space. In the Euclidean capillary setting, the capillary $C^2$64 dual Minkowski problem is reduced to a Monge–Ampère type equation with Robin boundary condition on the unit spherical cap, and there exists a unique smooth solution provided $C^2$65 (Gao, 3 Oct 2025). Since that result is formulated for Euclidean capillary hypersurfaces, its direct anisotropic analogue remains a separate question.

6. Flows, erosion, monotonicity, and analytical extensions

A major dynamical result is the convergence of volume-preserving anisotropic curvature flows to Wulff shapes. For a smooth closed strictly convex hypersurface $C^2$66, the flow

$C^2$67

with $C^2$68 chosen to preserve enclosed volume, exists for all time and converges in the Hausdorff sense to the Wulff shape. In the cases $C^2$69, $C^2$70, or $C^2$71, the Hausdorff convergence improves to smooth and exponential convergence (Andrews et al., 2021).

This flow result is not merely asymptotic. Its proof uses the curvature-measure rigidity theorem: after extracting a limit convex body $C^2$72, weak continuity of anisotropic curvature measures gives a relation of the form

$C^2$73

and the characterization theorem then identifies $C^2$74 as a scaled Wulff shape (Andrews et al., 2021). In that sense, anisotropic curvature measures provide both static classification and dynamic compactness.

A related erosion theory is provided by anisotropic inner parallel bodies. Given convex bodies $C^2$75, the inner parallel sets

$C^2$76

satisfy

$C^2$77

and the anisotropic isoperimetric quotient

$C^2$78

is nonincreasing, generically strictly decreasing. Equality characterizes homothety to tangential bodies of $C^2$79 (Crasta, 2021). This gives a complementary picture: Wulff geometry is the equilibrium of volume-preserving curvature relaxation, whereas tangential-body geometry governs self-similar anisotropic erosion.

Monotonicity under inclusion is another structural property with direct capillarity implications. If $C^2$80 is positively $C^2$81-homogeneous and convex, and $C^2$82 are convex bodies, then

$C^2$83

Stefani further proved a quantitative lower bound on the perimeter deficit $C^2$84 in terms of the Hausdorff distance and a critical cross-section of $C^2$85 (Stefani, 2016). This is the anisotropic version of the monotonicity of surface area for nested convex bodies.

That monotonicity is highly rigid. Among weighted isotropic perimeters, only constant multiples of the Euclidean perimeter satisfy monotonicity on nested convex bodies. Although the analogous result fails for general weighted anisotropic perimeters, an analogous characterization does hold for radially weighted anisotropic densities $C^2$86: if monotonicity holds for all nested convex bodies, then the radial weight $C^2$87 must be constant (Saracco et al., 2023). This rules out another common misconception: spatial inhomogeneity is not generically compatible with the inclusion-monotonicity that homogeneous anisotropic capillary energies enjoy.

Finally, anisotropic symmetrization provides the functional-analytic counterpart of Wulff rigidity. For a convex body $C^2$88 and anisotropic perimeter $C^2$89, the Wulff inequality identifies dilates of $C^2$90 as the unique volume-constrained minimizers. In the Sobolev setting, double symmetrization yields

$C^2$91

and equality forces level sets to be homothetic to the relevant Wulff-type convex bodies (Bianchi et al., 2024). This suggests that anisotropic capillary convex bodies are not only geometric equilibria of surface-tension problems but also the extremal level-set geometries of a wider class of anisotropic variational inequalities.