Willmore-type inequality in unbounded convex sets
Xiaohan Jia Guofang Wang Chao Xia Xuwen Zhang

	
	Willmore inequality]Willmore-type inequality in unbounded convex sets


Jia]Xiaohan Jia
	[X.J]School of Mathematics

Southeast University

 211189, Nanjing, P.R. China
	xhjia@seu.edu.cn
	XJ is  supported by the NSFC (Grant No. 12401249) and Natural Science Foundation of Jiangsu Province, China (Grant No. BK20241258)
	
		Wang]Guofang Wang
	[G.W]Mathematisches Institut

		Universität Freiburg

	Ernst-Zermelo-Str.1

		79104

  Freiburg
 Germany
	guofang.wang@math.uni-freiburg.de
 
Xia]Chao Xia
	[C.X]School of Mathematical Sciences

		Xiamen University

		361005
 Xiamen
 P.R. China
	chaoxia@xmu.edu.cn
 CX is  supported by the NSFC (Grant No. 12271449, 12126102)
 
	Zhang]Xuwen Zhang
	[X.Z]Mathematisches Institut

		Universität Freiburg

	Ernst-Zermelo-Str.1

		79104

  Freiburg
 Germany
xuwen.zhang@math.uni-freiburg.de
	
	
		
	
	§ ABSTRACT 

In this paper we prove the following Willmore-type inequality: On an unbounded closed convex set K⊂^n+1 ( n≥ 2), for any embedded hypersurface ⊂ K with boundary ⊂ K satisfying certain contact angle condition, there holds

1/n+1∫_H^n A≥AVR(K)^n+1.

Moreover, equality holds if and only if  is a part of a sphere and  K∖ is a part of the solid cone determined by . Here  is the bounded domain enclosed by  and K, H is the normalized
mean curvature of , and AVR(K) is the asymptotic volume ratio of K.
We also prove an anisotropic version of this Willmore-type inequality. 
As a special case, we obtain a Willmore-type inequality for anisotropic capillary hypersurfaces in a half-space.


		
				
		MSC 2020: 53C42, 53C20 .

		Keywords:   Willmore inequality, free boundary hypersurface, capillary hypersurface, anisotropic mean curvature, 
  asymptotic volume ratio

		
	


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§ INTRODUCTION


The classical Willmore inequality  Willmore68,Chen71 states that for a bounded domain ⊂^n+1 with smooth boundary, it holds that

1/n+1∫_H^n A≥^n+1
,

where H is the normalized mean curvature of  and ^n+1 is the volume of unit ball ^n+1.
Moreover, equality in (<ref>) holds if and only if  is a round ball.

Recently, Agostiniani-Fogagnolo-Mazzieri <cit.> proved the following Willmore-type inequality in Riemannian manifolds with nonnegative Ricci curvature.
[<cit.>*Theorem 1.1]
Let (M^n+1,g)  (n≥2) be a complete Riemannian manifold with nonnegative Ricci curvature and ⊂ M a bounded open set with smooth boundary.
Then

 1/n+1∫_
H^n A≥AVR(g)^n+1,


where AVR(g) is the asymptotic volume ratio of M. Moreover, if AVR(g)>0, equality holds if and only if M∖ is isometric to ([r_0,∞)×, r^2+(r/r_0g_)^2) with

r_0
=(/AVR(g)^n)^1/n.



Soon after Wang <cit.> gave a short proof of Theorem <ref>, which is based on Heintze-Karcher's comparison theorem in Riemannian geometry.

In this paper, we study a similar problem in Euclidean unbounded closed convex sets. Let us first introduce some terminologies and properties of unbounded closed convex sets, which are needed to state our main result.
Let K⊂^n+1 ( n≥2) be an unbounded closed convex set.
We denote by Reg( K) the regular part of K, that is, the set of points near which K can be locally written as a C^1-hypersurface, and  Sing( K)= K∖ Reg( K) the singular part of K.
For x∈ Reg( K), we denote by N̅(x) the outward unit normal to K at x. Let B_R(x) denote the open ball of radius R centered at x.




We define 
 AVR(K)
 =lim_R→∞ |B_R(0)∩K|/^n+1R^n+1.

For the well-definedness  of AVR(K) and its generalization see Appendix <ref> below. Moreover,
It is not difficult to check  that 

 AVR(K)
 =lim_R→∞ |B_R(p)∩K|/^n+1R^n+1,

for any p∈^n+1. 



Our main result in this paper is the following Willmore-type inequality in unbounded convex sets.


Let K⊂^n+1 be an unbounded closed convex set. 


Let ⊂K be a compact, embedded C^2-hypersurface with boundary ⊂ Reg( K) intersecting K transversally such that

	ν(x),N̅(x)≥0,   for any  x∈.

Here ν denotes the outward unit normal to  with respect to , the bounded domain enclosed by  and K.
Then there holds


1/n+1∫_H^n A≥AVR(K)^n+1.


Moreover, equality in (<ref>) holds if and only if  is a part of a sphere and 
 K∖ is a part of the solid cone determined by .






Our theorem can be viewed as an extrinsic counterpart of Theorem <ref>.
The approach to this theorem is inspired by Wang's short proof <cit.> of Theorem <ref> based on Heintze-Karcher's comparison <cit.>, and also inspired by our recent works on Heintze-Karcher-type inequalities in various circumstances, see JWXZ22,JWXZ23,JWXZ23b.

We refer the interested reader to <cit.>*Theorem 1.2 for the Heintze-Karcher-type inequality in arbitrary convex sets, which could be viewed as  a "dual" version of the above Willmore inequality in unbounded convex sets.


We would like to call attention to a result by Choe-Ghomi-Ritore <cit.>, which says that for a compact free boundary hypersurface ⊂^n+1 outside a convex set, it holds that
 
  1/n+1∫_H^n dA≥1/2 ^n+1 with equality holds if and only if  is a hemisphere lying in a half-space. This inequality leads to an optimal relative  isoperimetric inequality in <cit.>. See also <cit.>. 
In contrast all our results in this paper hold for compact hypersurfaces in a convex set.


 

In view of our previous work <cit.>, it is not surprising that we could in fact establish the following Willmore-type inequality in unbounded convex sets, with anisotropy taken into account. Here anisotropy means a smooth positive function F: 𝕊^n→ℝ_+ on the unit sphere (𝕊^n,σ) such that 
(∇^2 F+F σ) is positive definite.  In literature F is usually called a Minkowski norm. (F induces a norm if F is even, i.e, F(-x)=F(x). In the paper we do not require the evenness.)

With respect to a Minkowski norm F,
one can define 
(unit) Wulff ball ^F,  asymptotic volume ratio AVR_F for unbounded convex sets, the anisotropic unit normal ν_F and the normalized anisotropic mean curvature H^F for a hypersurface. For more definitions and notation see

Section <ref> and also Appendix <ref> below.

Let K⊂^n+1 be an unbounded closed convex set. 
Let ⊂K be a compact, embedded C^2-hypersurface with boundary ⊂ Reg( K) intersecting K transversally such that

	ν_F(x),N̅(x)≥0,   for any  x∈.

Then there holds


1/n+1∫_F(ν)H^F^n A≥AVR_F(K)^F.


Moreover, equality in (<ref>) holds if and only if  is a part of a Wulff shape and K∖ is  a part of the solid cone determined by .

Note that Theorem <ref> is a special case of Theorem <ref> with F(ξ)=ξ.

The Willmore-type inequality (<ref>) may be applied to prove relative isoperimetric inequality in unbounded convex sets, as Choe-Ghomi-Ritore <cit.> did outside convex sets. We remark that the relative isoperimetric-type inequality in unbounded convex sets in the isotropic case, i.e. F(ξ)=|ξ|, has been proved by  Leonardi-Ritoré-Vernadakis <cit.>. 


When K=^n+1_+, the upper half-space, by choosing F(ξ)=|ξ|-cosθ_0 ξ, E_n+1 for some θ_0∈ (0, π), we get from Theorem <ref> the following Willmore-type inequality for capillary hypersurfaces.

Given θ_0∈(0,π).
Let  ⊂_+^n+1 be a compact, embedded C^2-hypersurface with boundary ⊂_+^n+1 intersecting ^n+1_+ transversally such that

<ν, -E_n+1>≥-cosθ_0,   for any  x∈.

Then there holds

1/n+1∫_Σ(1-cosθ_0ν, E_n+1)H^n A≥B_1,θ_0,

where  
B_1,θ_0=B_1(-cosθ_0 E_n+1)∩_+^n+1.
Moreover, equality in (<ref>) holds if and only if  is a θ_0-capillary spherical cap in _+^n+1.

More generally, by choosing F(ξ) in Theorem <ref> to be F(ξ)+_0<ξ,E^F_n+1> for some ω_0, we get the following Willmore-type inequality for anisotropic capillary hypersurfaces.



Given _0∈(-F(E_n+1),F(-E_n+1)).
Let  ⊂^n+1_+ be a compact, embedded C^2-hypersurface with boundary ⊂_+^n+1 intersecting ^n+1_+ transversally such that

  ν_F(x), -E_n+1=(x)≥_0,   for  any x∈∂Σ.

Then there holds

1/n+1∫_(F(ν)+_0<ν,E^F_n+1>)H^F^nA≥^F_1,_0,

where ^F_1,_0=^F_1(_0 E^F_n+1)∩_+^n+1.

Moreover, equality in (<ref>) holds if and only if  is an anisotropic _0-capillary Wulff shape in _+^n+1.




In fact, as already observed in De Phillipis-Maggi's work <cit.>, anisotropic capillary problems with respect to F can be regarded as anisotropic free boundary problems with respect to another Minkowski norm F̃ in the half-space.





Hypersurfaces with the aforementioned special boundary conditions naturally arise from Calculus of Variations and are nowadays of particular interest.
We refer the readers to <cit.> and the references therein for a short historical introduction.

For the specialty of the half-space, we could in fact prove Theorem <ref> in the half-space case in an alternative way.
This is done by a geometric observation on the Gauss image of , stated in Proposition <ref>.
Once this is established, the rest of the proof of the Willmore inequality in the half-space then follows from the classical argument based on an area formula.







The rest of the paper is organized as follows. 
In Section <ref>, we collect basic knowledges on Minkowski norm, anisotropic geometry, and anisotropic capillary hypersurfaces.
In Section <ref>, we prove Theorem <ref>.
In Section <ref>, we give an alternative proof for Theorem <ref> in the half-space case and then apply it to anisotropic capillary hypersurfaces.







§ PRELIMINARIES






 §.§ Notations


The Euclidean metric, scalar product, and Levi-Civita connection of the Euclidean space ^n+1 are denoted respectively by g_ euc,<·,·>, and D.
When considering the topology of ^n+1, we adopt the following notations:
for a set E⊂^n+1,
we denote
by E the topological closure of E, by int(E) the topological interior of E, and by E the topological boundary of E in ^n+1.
Regarding the use of the symbol ·,
if we plug in a vector e∈^n+1, then e denotes the Euclidean length of e.
If we plug in a k-dimensional submanifold M⊂^n+1, then we write

M
ℋ^k(M),

where ℋ^k is the k-dimensional Hausdorff measure in ^n+1.
In particular, if we plug in an open set  of ^n+1, then we mean


ℒ^n+1().

To avoid ambiguity, we also use Vol(·) to denote the ^n+1-measure of certain sets in ^n+1, see for example (<ref>).







 §.§ Minkowski norm and anisotropic geometry

Let F: 𝕊^n→ℝ_+ be a smooth positive function on the standard sphere (𝕊^n,σ) such that 

A_F
∇^2 F+F σ>0,

where  denote the Levi-Civita connection corresponding to σ.



A Minkowski norm is the one homogeneous extension of any such F to the whole ^n+1, namely,
F(ξ)=|ξ|F(ξ/|ξ|) for ξ≠ 0 and F(0)=0.
Note that condition (<ref>) is equivalent to saying that 1/2F^2 is uniformly convex, in the sense that

D^2(1/2F^2)(ξ)>0,  ∀ξ∈^n+1.
 




Let Φ be the Cahn-Hoffman map associated to F, which is given by

Φ(z)=DF(z)=∇F(z)+F(z)z,   ∀z ∈𝕊^n.

The image Φ(𝕊^n) of Φ  is called (unit) Wulff shape.

The dual Minkowski norm of F, denoted by F^o, is given by

    F^o(x)=sup{<x,z>/F(z)|   z∈^n}.


We collect some well-known facts on F, F^o, and Φ, see e.g., <cit.>*Proposition 2.1.
 For any z∈^n+1∖{0}
the following statements hold for an Minkowski norm.

    
  (i) F^o(tz)= t F^o(z) , for any t>0.
    
  (ii) F^o(x+y)≤ F^o(x)+F^o(y), for x, y ∈^n+1.

  (iii) <Φ(z),z>=F(z).

  (iv) F^o(Φ(z))=1.

  (v) The following Cauchy-Schwarz inequality holds:

    z, ξ≤ F^o(z)F(ξ),   ∀ξ∈^n+1.


  (vi)  The  Wulff shape Φ(𝕊^n)={x∈^n+1|F^o(x)=1}.


We denote the Wulff ball of radius r and centered at x_0∈^n+1  by

_r^F(x_0)
={x∈^n+1|F^o(x-x_0)<r},

and the corresponding Wulff shape _r^F(x_0) is given by

_r^F(x_0)
={x∈^n+1|F^o(x-x_0)=r}.

We also use ^F and ^F_R to abbreviate  ^F_1(0) and ^F_R(0). 



Let ⊂^n+1 be a C^2-hypersurface and ν  a unit normal field of .
The anisotropic normal of  with respect to ν  and F is given by

ν_F=Φ(ν)=∇F(ν)+F(ν)ν,

and the anisotropic principal curvatures {κ_i^F}_i=1^n of  with respect to ν and F are given by the eigenvalues of the anisotropic Weingarten map 
    ν_F=A_F(ν)∘ν: T_p→ T_p.
 The eigenvalues are real since A_F is positive definite and symmetric.
Let 
    H^F=1/n∑_i=1^nκ_i^F    and    H^F_n=∏_i=1^nκ_i^F
 denote the normalized anisotropic mean curvature and the anisotropic Gauss-Kronecker curvature of Σ respectively.




It is easy to check that the anisotropic principal curvatures of ^F_r(x_0) are 1/r, since 

    ν_F(x)=x-x_0/r,     on ^F_r(x_0).

We record
the following very useful anisotropic angle comparison principle.

Let x,z∈^n be two distinct points and y∈^n lie in a length-minimizing geodesic joining x and z in ^n,
then we have 
<Φ(x),z>≤<Φ(y),z>.

Equality holds if and only if x=y.






 §.§ Anisotropic capillary hypersurfaces in a half-space


Let us first recall the definition of anisotropic capillary hypersurface in the half-space.

Given _0∈(-F(E_n+1),F(-E_n+1)), any hypersurface ⊂^n+1_+ is said to be anisotropic _0-capillary in ^n+1_+ if it intersects ^n+1_+ transversally with

<ν_F(x),-E_n+1>≡_0,∀x∈.

In particular,  is said to be anisotropic free boundary  if it is anisotropic _0-capillary with _0=0.

Here we record some facts concerning anisotropic capillary hypersurfaces in the half-space.


Let  be a hypersurface in ^n+1_+ which meets ^n+1_+ transversally
and define a function on ∂Σ by (x)<ν_F(x),-E_n+1>. Then for any x∈,

(x)∈(-F(E_n+1),F(-E_n+1)).



Define a constant vector E^F_n+1∈ℝ^n+1 as in <cit.>,

E^F_n+1
={
[ -Φ(-E_n+1)/F(-E_n+1),             if  _0>0,;    Φ(E_n+1)/F(E_n+1),             if  _0<0,;                E_n+1,             if  _0=0,;                       ]
.

whose definition is strongly related to the Cauchy-Schwarz inequality.
When _0=0, one can also define E_n+1^F as Φ(E_n+1)/F(E_n+1) or -Φ(-E_n+1)/F(-E_n+1).
Note that E^F_n+1, E_n+1=1, and when F is the Euclidean norm, E^F_n+1 is indeed E_n+1.


For _0∈(-F(E_n+1),F(-E_n+1)), there holds

    F(z)+_0z,E^F_n+1>0,  for any z∈^n.



We have the following integral formula.




Let  be a compact, embedded C^2-hypersurface in ^n+1_+ which intersects ^n+1_+ transversally.
Denote by  the bounded domain enclosed by  and ^n+1_+.
Then
there holds

∫_ν,E_n+1^F A
=∫_∩^n+1_+ A.



Notice that div(E_n+1^F)=0.
Integrating this over , using integration by parts, then invoking the fact that <E_n+1,E^F_n+1>=1, we obtain the assertion.


As mentioned in the introduction, we shall interpret the anisotropic capillary problem as an anisotropic free boundary problem.
This is done by introducing the following Minkowski norm.

Given _0∈(-F(E_n+1),F(-E_n+1)).
Let  be a compact, embedded C^2 _0-capillary hypersurface in ^n+1_+.
Then  is  an anisotropic free boundary hypersurface in ^n+1_+ with respect to the Minkowski norm F̃, defined by

F̃(ξ)
F(ξ)+_0ξ,E_n+1^F,  ∀ξ∈^n+1.

Moreover, 

    
  * A_F=A_F̃ on Σ, and hence the anisotropic curvatures of Σ w.r.t. to F and F̃ are the same.
    
  * The unit Wulff shapes of associated to F and F̅ are the same up to a translation, precisely, 
    ^F̃=_1^F(_0E_n+1^F).




By direct computation, we see that

DF̃(ξ)
=D F(ξ)+_0E_n+1^F.

It follows that for every p∈⊂^n+1_+, there holds

<ν_F̃(p),-E_n+1>
=<ν_F(p)+_0E_n+1^F,-E_n+1>
=_0-_0=0,

where we have used the fact that <E_n+1,E^F_n+1>=1.
This proves the first part of the assertion.

It is direct to see that A_F(p)=A_F̃(p) for any p∈, since we have

D^2F̃(ξ)
=D^2F(ξ).

The last part of the assertion follows from (<ref>), and the fact that the unit Wulff shape with origin as its center could be characterized by ^F̃=DF̃(^n).


Since  F is in general not an even function on ^n, for later purpose, we need the following  Minkowski norm. Define F_∗  by 

    F_∗(z)=F(-z),    z∈^n.

Geometrically speaking, F_∗ is induced by the convex body which is centrally symmetric to ^F. It is clear that F_∗ also induces a smooth Minkowski norm, still denote by F_∗, and we 
denote by Φ_∗ and F_∗^o the Cahn-Hoffman map and the dual Minkowski norm associated to F_∗. 

There hold that

    
  * 
    F_∗^o(x)   =F^o(-x),  ∀x∈^n+1.
    
    
  * 
Φ_∗(z)   =-Φ(-z),  ∀z∈^n.

     
     
   
     
  * For any x∈,
     let {κ_i^F(x)}_i=1^n denote the  anisotropic principal curvatures of  at x with respect to ν and F.
     Then {-κ_i^F(x)}_i=1^n
     are the anisotropic principal curvatures of  at x with respect to the unit inner normal -ν and F_∗,  which we denote by {κ_i^F_∗(x)}_i=1^n.
     
     
     



To prove (<ref>), we verify by definition.
Precisely, thanks to (<ref>), we have

F^o_∗(x)
=sup{<x,z>/F_∗(z)|   z∈^n}
=   sup{<x,z>/F(-z)|   z∈^n}

=   sup{<-x,-z>/F(-z)|   -z∈^n}
=F^o(-x).

(<ref>) follows directly by
differentiating both sides of the equality F_∗(z)=F(-z).

To prove (3), we fix any x∈ and let {e_i(x)}_i=1^n denote the anisotropic principal vector at x∈ corresponding to κ_i^F, that is,

D_e_i(x)Φ(ν(x))
=κ^F_i(x)e_i(x).

Letting z=-ν(x) in (<ref>) then
differentiating with respect to e_i(x), we obtain

D_e_i(x)Φ_∗(-ν(x))
=-D_e_i(x)Φ(ν(x))
=-κ^F_i(x)e_i(x),

thereby showing that {e_i(x)}_i=1^n are also the anisotropic principal vectors at x∈, corresponding to {κ_i^F_∗(x)}_i=1^n, with

κ_i^F_∗(x)
=-κ_i^F(x),  ∀i=1,…,n.

This completes the proof.


We close this subsection by stating the following geometric result, which follows
simply by using integration by parts for the divergence of the position vector field.




We have

∫_^F∩^n+1_+F(ν) A
=(n+1)^F∩^n+1_+
and

∫_^F_1,_0∩^n+1_+(F(ν)+_0<ν,E_n+1^F>)A
=(n+1)_1,_0^F,

where ^F_1,_0=^F_1(_0 E^F_n+1)∩_+^n+1.


     (<ref>) follows from integration by parts of the divergence of the position vector field. (<ref>)
follows from     plugging F̃ given by (<ref>) into (<ref>).



















§ WILLMORE INEQUALITY IN UNBOUNDED CONVEX SETS


As in <cit.>, we introduce the following flow which generates parallel hypersurfaces in the anisotropic free boundary sense, defined by

ζ_F(x,t)
=x+tΦ(ν(x)), (x,t)∈×_+ Z.

In the rest of the section,
we write ^F_r,K=^F_r(0)∩K for any r>0.


Given an unbounded, closed convex sets K  in ^n+1.
Let ⊂ K be a compact, embedded C^2-hypersurface with boundary ⊂ Reg( K) such that (<ref>) holds.
Denote by  the bounded domain delimited by  and K.
 Assume that 0∈ int(∩ K).
Define a distance function on K by 

d_F_∗(y,Ω)=inf_r>0{r|^F_∗_r(y)∩Ω≠∅}.

Then we have following statements.

    
  * For any R>0, there holds

   {y∈K∖Ω:d_F_∗(y,Ω) ≤R}

⊂   {ζ_F(x,t)|x∈Σ,t∈(0,min(R,τ(x))]},

where
τ is a function defined on  by 

τ(x)=

+∞,    if κ_i^F_∗(x)≤0  for any  i=1,⋯,n,

1/max_i κ_i^F_∗(x),   otherwise.


    
  * For any R>0, there holds

{  y∈K:d_F_∗(  y,^F_r,K) ≤R}
=^F_r+R,K.

    
  * 
 lim_R→+∞Vol({  y∈K:d_F_∗(  y,) ≤R})/^FR^n+1
 
 =AVR_F(K),
 
 where AVR_F(K) is defined and discussed in Appendix <ref> below.



(1)





For any y∈K∖Ω satisfying d_F_∗(y,Ω)≤ R, we use a family of closed Wulff balls {^F_∗_r(y)}_r>0 to touch Ω.
Clearly there must exist x∈∂Ω and 0<r_0≤ R,
such that  _r_0^F_∗(y) touches Ω for the first time at a point x.
Let ν^∗(x) be  the outer unit normal of _r_0^F_∗(y) at x, and Φ_∗(ν^∗(x)) the anisotropic normal at x satisfying 

Φ_∗(ν^∗(x))
=x-y/r_0.

By convexity of K and strictly convexity of _r_0^F_∗(y),  x cannot be obtained at ∂Ω∖Σ, thus only the following two cases are possible:

Case 1.  x∈∂Σ.

Since ^F_∗_r_0(y) touches Ω from outside at x, ν^∗(x), -ν(x) and -N̅(x) lie on the same 2-plane and moreover, ν^∗ lies in a length-minimizing geodesic on ^1 joining -ν and -N̅(x).
Thanks to Proposition <ref>, we find

Φ_∗(ν^∗(x)),-N̅(x)
   ≥Φ_∗(-ν(x)),-N̅(x)

   =-Φ(ν(x)),-N̅(x)
=ν_F(x),N̅(x)
≥0.

On the other hand, since K is convex and y∈ K, we must have

0
≥y-x,N̅(x)=-r_0Φ_∗(ν^∗(x)), N̅(x)
= r_0Φ_∗(ν^∗(x)),-N̅(x).

 Hence, the above two inequalities must hold as equalities simultaneously; that is to say, x must belong to {x∈∂Σ|ν_F(x),N̅(x)=0}, while ν^∗(x)=-ν(x), then there hold that

y=x-r_0Φ_∗(-ν) 
=x+r_0Φ(ν)
=ζ_F(x,r_0),

and that Ω and   ^F_∗_r_0(y) are mutually tangent at x.
 Recall that κ_i^F_∗(x) denotes the anisotropic principal curvatures of  at x, with respect to -ν and F_∗, we therefore find max_iκ_i^F_∗(x)≤1/r_0.
Taking the definition of τ(x) into account, this readily implies r_0≤τ(x).
 


Case 2. x∈ int(Σ).

In this case, it is easy to see that ν^∗(x)=-ν(x) from the first touching property. We may conduct a similar 
argument as above to find that y can be written as y=ζ_F(x,r_0), with r_0≤τ(x).






Therefore, for any y∈K∖Ω, if  d_F_∗(y,Ω)≤ R, we can find x∈Σ and r_0∈(0,min(R,τ(x))], such that y=ζ_F(x,r_0), which implies  (<ref>).


(2) Now we show (<ref>).
Recall that we have set ^F_r,K=^F_r(0)∩ K.

Our first observation is that, 
if y∈^F_r,K, it is easy to see that d_F_∗( y,^F_r,K)=0.
Since it is trivial to see that ^F_r,K⊂^F_r+R,K, to prove (<ref>), it suffices to show that

  {  y∈K∖^F_r,K:d_F_∗(y,^F_r,K) ≤R}=^F_r+R,K∖^F_r,K,   ∀R>0.


Claim.
For any y∈K∖^F_r,K,
let R̃=d_F_∗(y,^F_r,K),
then F^o(y)=r+R̃.

To prove the claim,
we consider the family of closed Wulff balls {^F_∗_r(y)}_r>0, as r increases, by the definition of d_F_∗, we have that
_R̃^F_∗(y) touches ^F_r,K for the first time. 
 
Suppose that  _R̃^F_∗(y) touches ^F_r,K at x, then
x∈∂^F_∗_R̃(y)∩∂^F_r,K.

Since
_R̃^F_∗(y) is strictly convex, 
and K is convex,

we deduce that 
  

x∈∂^F_∗_R̃(y)∩∂^F_r(0)∩K.

Let ν(x), ν̃(x) denote respectively the unit outer normals of ^F_∗_R̃(y) and ^F_r(0) at x.

It follows that
  
 x-y=R̃ Φ_∗(ν(x)),  x=rΦ(ν̃(x)).
 
 
If x∈∂^F_∗_R̃(y)∩∂^F_r(0)∩∂ K, we have -ν(x) lies in a length-minimizing geodesic joining ν̃(x) and N̅(x) in ^n.
 By Proposition <ref>, it holds that

Φ(ν̃(x)),N̅(x)≤Φ(-ν(x)), N̅(x)=-Φ_∗(ν(x)),N̅(x).

On the other hand, it follows from (<ref>)  and convexity of K that

Φ(ν̃(x)),N̅(x)
=x/r,N̅(x)≥0,

-Φ_∗(ν(x)),N̅(x)
=-x-y/R̃,N̅(x)
≤0.

Combining all above,
we thus obtain ν(x)=-ν̃(x) and
x,N̅(x)=<y,N̅(x)>=0.

If x∈∂^F_∗_R̃(y)∩∂^F_r(0)∩ int(K), it is easy to see that  ν(x)=-ν̃(x).
 
Summarizing, in either case, we always have ν(x)=-ν̃(x), thus

Φ_∗(ν)=-Φ(-ν)=-Φ(ν̃).

Using (<ref>) again, we get 
 
 
 
 

F^o(y)
=   F^o(y-x +x)

=   F^o(-R̃Φ_∗(ν(x))+rΦ(ν̃(x)))

=   F^o((r+R̃)Φ(ν̃(x)))
=r+R̃,

which proves the claim.
Now we prove (<ref>).
 
"⊂": For any y∈  K∖^F_r,K satisfying R̃=d_F_∗(  y,^F_r,K)≤ R, we deduce immediately from the above estimate that
 










F^o(y)
= r+R̃
≤r+R,

thus y∈^F_r+R,K∖^F_r,K.
 
"⊃": For any y∈^F_r+R,K∖^F_r,K, suppose that R̃=d_F_∗(y,^F_r,K)> R, then it holds that

F^o(y)
=r+R̃
>r+R,

which contradicts to y∈^F_r+R,K. Hence d_F_∗(y,^F_r,K)≤ R, and the proof is completed.

(3)
Since Ω is bounded, we could find r_1 and r_2 such that ^F_r_1,K⊂Ω⊂^F_r_2,K. From the definition of d_F_∗ (<ref>), we find 

d_F_∗(y,^F_r_2,K)≤d_F_∗(y,)≤d_F_∗(y, ^F_r_1,K),
  ∀y∈K,

and it follows that
 

    {  y∈K:d_F_∗(  y,^F_r_1,K)≤R}

⊂   {  y∈K:d_F_∗(  y,Ω) ≤R}

 ⊂   {  y∈K:d_F_∗(  y,^F_r_2,K) ≤R}.
 

Dividing (<ref>) by ^FR^n+1, we get (<ref>)  by using (<ref>) and Proposition <ref>.



We have now all the requisites to prove the Willmore inequality.


 



 



We first prove the Willmore inequality.
Our starting point is, thanks to (<ref>), we may use the area formula to estimate the volume as follows: for any R>0,

Vol({  y∈K:d_F_∗(  y,Ω)
≤R})
≤|Ω|+∫_Σ∫_0^min(  R,τ(  x))J^Zζ_F(x,t)tA.

By a simple computation, we see, the tangential Jacobian of ζ_F along Z=×_+ at (x,t) is just

    J^Zζ_F(x,t)
    = F(ν(x))∏_i=1^n(1+tκ_i^F(x)).

Recall the definition of τ (<ref>), and taking also Proposition <ref>(3) into account, 
we may rewrite τ as

τ(x)=

+∞,    if κ_i^F(x)≥0  for any  i=1,⋯,n,

-1/min_i κ_i^F(x),   otherwise.



 


 It is clear that for any x∈, there hold 1+tκ_i^F(x)>0, for each i=1,⋯,n, and for any t∈(0,τ(x)).
By the

AM-GM inequality, 
we have 

J^Zζ_F(x,t)
≤F(ν)(1+H^F(x)t)^n.


To have a closer look at (<ref>), we divide  into two parts _+={x∈Σ:H^F(x)>0} and ∖Σ_+. 
On ∖_+, we have 

0≤(1+H^Ft)^n≤1, ∀t∈[0,τ(x)],

which, in conjunction with (<ref>), gives

   ∫_∖_+∫_0^min(R,τ(x))J^Zζ_F(x,t)tA

≤O(R).

Thus (<ref>) can be further estimated as follows:

   Vol({y∈K:d_F_∗(  y,)≤R})

≤   
+∫__+∫_0^min(  R,τ(x))J^Zζ_FtA 
+∫_∖_+∫_0^min(R,τ(x))J^Zζ_FtA


≤   ∫__+∫_0^RF(ν)(1+H^F(x)t)^ntA+O(R)

=   R^n+1/n+1∫__+F(ν)(H^F(x))^nA
+O(R^n)

≤   R^n+1/n+1∫_F(ν)H^F^nA
+O(R^n),

where the third inequality holds due to (<ref>).
 

Dividing  both sides of (<ref>) by R^n+1 and
letting R→ +∞, we deduce from (<ref>) that 

AVR_F(K)^F
≤1/n+1∫__+F(ν)(H^F)^nA ≤1/n+1∫_F(ν)H^F^nA.

which finishes the proof of the inequality (<ref>).



Now we start to prove  the rigidity. First we claim



Claim 1. 
If the equality in (<ref>) holds, 

then H^F≥0 on ,
 τ(x)=+∞ on _+, and equality in (<ref>) holds  on _+.
 


To prove the first assertion of the claim,
we deduce from (<ref>) and the equality case of (<ref>) that

1/n+1∫_F(ν)H^F^nA
= 1/n+1∫__+F(ν)(H^F)^nA,

which implies H^F≥ 0 on .



To prove τ(x)=+∞ on _+, 
we argue by contradiction and
assume that there exists a point x_0∈_+ satisfying τ(x_0)<+∞. Since ∈ C^2, we can find a neighborhood of x_0 in _+, denoted by U(x_0), such that τ(x)≤ 2τ(x_0)<+∞ on U(x_0).  For any R>2τ(x_0), we obtain from (<ref>) that

   Vol({y∈K:d_F_∗(  y,)≤R})

≤    ∫__+∫_0^min(R,τ(x))F(ν)(1+H^F(x)t)^ntA +O(R)



≤    ∫__+∖U(x_0)∫_0^RF(ν)(1+H^F(x)t)^ntA 

   +∫_ U(x_0)∫_0^τ(x)F(ν)(1+H^F(x)t)^ntA +O(R)

≤   ∫__+∖U(x_0)∫_0^RF(ν)(1+H^F(x)t)^ntA  +O(R)

=   R^n+1/n+1∫_Σ_+∖U(x_0) F(ν)(H^F)^nA +O(R^n).

Dividing  both sides by R^n+1 and
letting R→ +∞, it follows that 

AVR_F(K)^F
≤1/n+1∫__+∖U(x_0)F(ν)(H^F)^nA,

which is a contradiction to the assumption that equality in (<ref>) holds. 

By a similar contradiction argument, we also get that
equality in (<ref>) holds  on _+, and we omit the proof here for brevity. 
In particular, this proves Claim 1.





 
In virtue of this claim, we see that  =_+ must be an anisotropic umbilical hypersurface, and hence a part of a Wulff shape.
Write =_R_0^F(x_0)∩ K for some R_0>0 and x_0∈^n+1.
Clearly,  determines an solid cone with vertex at x_0, given by

_x_0
={x_0+tx-x_0/R_0:x∈int(), t∈(0,+∞)}.


Consider now the modified convex set 
    K=(K∖)∪(_x_0∩_R_0^F(x_0)).

Due to convexity of K and the boundary condition (<ref>), for any x∈∂Σ, there exists at least one supporting hyperplane passing through x, hence K is also convex.

Claim 2. ^F_R(x_0)∩K/R^n+1 is non-increasing on R∈ [0,+∞).

This can be proved similarly as Proposition <ref> below.
Indeed,
by using the co-area formula, we have 

/R|^F_R(x_0)∩K|/R^n+1
=∫_^F_R(x_0)∩K1/|DF^o(x-x_0)| A- (n+1)|^F_R(x_0)∩K|/R/R^n+1.

On the other hand, by the divergence theorem, we have for R≥ R_0,

(n+1)|^F_R(x_0)∩K|=   ∫_^F_R(x_0)∩Kdiv(x-x_0)x 

=   ∫_^F_R(x_0)∩KR/DF^o(x-x_0)A

   +∫_^F_R(x_0)∩K∖Ωx-x_0,N̅(x)A

   +∫__x_0∩^F_R_0(x_0)x-x_0,N̅(x)A.

Note that x-x_0,N̅(x)=0 on _x_0∩^F_R_0(x_0) since it is part of the boundary of a cone, and x-x_0,N̅(x)≥0 on ^F_R(x_0)∩ K∖Ω since K is convex. 
It follows that  

(n+1)|^F_R(x_0)∩K|≥   R∫_^F_R(x_0)∩K1/DF^o(x-x_0)A, 

and in turn 
    / R|^F_R(x_0)∩K|/R^n+1≤ 0,

which proves the claim.

From the second claim, we know that 

AVR_F(K)^F
=   lim_R→+∞|^F_R(x_0)∩K|/R^n+1

≤   |^F_R_0(x_0)∩K|/R_0^n+1
=|^F_R_0(x_0)∩_x_0|/R_0^n+1.

On the other hand, 
using  equality in (<ref>) and the fact that  is anisotropic umbilical, we get

AVR_F(K)^F=   1/n+11/R_0^n∫__R_0^F(x_0)∩KF(ν) A
=   1/n+11/R_0^n∫__R_0^F(x_0)∩_x_0F(ν) A

=   |^F_R_0(x_0)∩_x_0|/R_0^n+1,

which 
implies that |^F_R(x_0)∩K|/R^n+1 is a constant for R∈[R_0,+∞).








See Fig. <ref> for an illustration when F is the Euclidean norm, where the red part denotes the boundary of the modified set K and the blue part denotes the boundary of the cone _x_0.
Because of the boundary condition (<ref>), we point out that Fig. <ref> is what we could expect so far.
Next, we are going to show by virtue of the claim that K is in fact _x_0, namely, the blue portion and the red portion in Fig. <ref> actually coincide.
In fact,
from the proof of the claim, we see that

    x-x_0,N̅(x)=0  on  K∖Ω,

which implies that K∖Ω is a part of boundary of the cone centered at x_0. Finally from the information that
 is a part of Wulff shape centered at x_0 and K∖Ω is part of boundary of the cone centered at x_0, we see that  is indeed an anisotropic free boundary Wulff shape in K, i.e., ν_F,N̅=0 along .
This completes the proof.











§ WILLMORE INEQUALITIES IN A HALF-SPACE



 §.§ An alternative approach in the half-space case
In this section, we use an alternative approach to prove Theorem <ref> in the half-space case. We restate it here.

Let ⊂^n+1_+ be a compact, embedded, C^2-hypersurface with boundary ⊂_+^n+1 such that

  ν_F(x), -E_n+1≥0,   for  any x∈∂Σ.

Then we have

1/n+1∫_ F(ν)H^F^n A≥|^F∩_+^n+1|.

Equality in (<ref>) holds if and only if  is an anisotropic free boundary Wulff shape.

The alternative approach is based on  the following geometric observation for the Gauss image of .

Let ⊂^n+1_+ be a compact, embedded, C^2-hypersurface with boundary ⊂_+^n+1 such that

  ν_F(x), -E_n+1≥0,   for  any x∈∂Σ.

Then the anisotropic Gauss map ν_F:^F satisfies

(^F∩_+^n+1)
⊂ν_F(_+).

where _+ is the subset of  where the anisotropic Weingarten map ν_F is positive-semi definite.





For any y∈^F∩^n+1_+, denote by N(y) the outer unit normal to ^F at y. Then Φ(N(y))=y.
Let _y be a closed half-space whose inward unit normal is given by N(y), such that ∩_y=∅, and let Π_y be the boundary of _y, which is a hyperplane in ^n+1.
Parallel translating Π_y towards  and denote by Π_y the hyperplane that touches  for the first time at some p∈.
If p∈⊂^n+1_+, then by the first touching property, we have
ν(p), -E_n+1≤N(y), -E_n+1.

Thanks to the first touching property, ν(p),N(y), and -E_n+1 are on the same 2-plane and from the above relation we know, N(y) lies in the length-minimizing geodesic on ^n joining ν(p) and -E_n+1.
Therefore using the anisotropic angle comparison principle (Proposition <ref>), we obtain

<ν_F(p),-E_n+1>
≤<Φ(N(y)),-E_n+1>= <y,-E_n+1>.

On the other hand, by assumption,
ν_F(p), -E_n+1
≥0
>y, -E_n+1,

which gives a contradiction.
Hence the first touching point p∈ int() and  is again tangent to Π_y at p. Thus
ν(p)=N(y) and ν_F(p)=Φ(ν(p))=y.
Note that the touching of  with Π_y is from the interior, therefore at the touching point p, ν(p)≥ 0 and in turn, ν_F(p)≥ 0.
Thus we have p∈_+ and ν_F(p)=y, which completes the proof.



Our starting point is Proposition <ref>.
To proceed, note that the Jacobian of the Gauss map with respect to  is just J^ν_F= H_n^F,  where H^F_n is the anisotropic Gauss-Kronecker curvature. By using (<ref>), the area formula and AM-GM inequality, we have

(n+1)^F∩^n+1_+
   =∫_^F∩^n+1_+F(ν(y))A(y)
≤∫__+F(ν(p))H_n^F(p) A(p)

   ≤∫__+F(ν)(H^F)^n A
≤∫_F(ν)H^F^n A,

which is (<ref>).

If equality in (<ref>) holds, then all the inequalities in the above argument are actually equalities, so we readily infer that  agrees with _+ 
and is in fact an anisotropic umbilical hypersurface, thanks to the AM-GM inequality. It is not hard to deduce that <ν_F(x),-E_n+1>=0. Hence  is an anisotropic free boundary Wulff shape in ^n+1_+.



















 §.§ Capillary hypersurfaces in a half-space


As corollaries of Theorem <ref>, we may deduce various Willmore-type inequalities for anisotropic capillary hypersurfaces in a half-space.

First, through the new Minkowski norm F̃ in (<ref>)  (see Proposition <ref>), the anisotropic capillary problem can be rephrased as an anisotropic free boundary problem.
Second, a special Minkowski norm  to our interest is defined as

F(ξ)=ξ-cosθ_0<ξ,E_n+1>,

for some angle constant θ_0∈(0,π), by virtue of which the capillary problem in a half-space could be interpreted in terms of anisotropic terminology (see e.g., DeMasi22,LXX23).
Plugging these Minkowski norms into Theorem <ref>, we thereby obtain the following Willmore inequalities as claimed in the introduction, which
we state for readers' convenience.


Given _0∈(-F(E_n+1),F(-E_n+1)).
Let  ⊂^n+1_+ be a compact, embedded, C^2-hypersurface with boundary ⊂_+^n+1 intersecting ^n+1_+ transversally such that

  ν_F(x), -E_n+1=(x)≥_0,   for  any x∈∂Σ.

Then there holds

1/n+1∫_(F(ν)+_0<ν,E^F_n+1>)H^F^nA≥^F_1,_0,

where ^F_1,_0=^F_1(_0 E^F_n+1)∩_+^n+1.

Moreover, equality in (<ref>) holds if and only if  is an anisotropic _0-capillary Wulff shape in ^n+1_+.



Given θ_0∈(0,π).
Let  ⊂_+^n+1 be a compact, embedded C^2-hypersurface with boundary ⊂_+^n+1 intersecting ^n+1_+ transversally, such that

<ν, -E_n+1>≥-cosθ_0,   for any  x∈.

Then there holds

1/n+1∫_Σ(1-cosθ_0ν, E_n+1)H^n A≥B_1,θ_0,

where  
B_1,θ_0=B_1(-cosθ_0 E_n+1)∩_+^n+1.
Moreover, equality in (<ref>) holds if and only if  is a θ_0-capillary spherical cap in ^n+1_+.


On the other hand, by revisiting the geometric property concerning Gauss map in the (anisotropic) capillary settings, we could prove a variant of Theorem <ref> as follows.

Given _0∈(-F(E_n+1),F(-E_n+1)).
Let  ⊂^n+1_+ be a compact, embedded, C^2-hypersurface with boundary ⊂_+^n+1 intersecting ^n+1_+ transversally such that

  ν_F(x), -E_n+1=(x)≥_0,   for  any x∈∂Σ.

Then there holds

∫_F(ν)H^F^n A≥∫_^F_1,_0∩^n+1_+F(ν)A.


Moreover, equality in (<ref>) holds if and only if  is an anisotropic _0-capillary Wulff shape in ^n+1_+.

The revisited variant of Proposition <ref> reads as follows.

Given _0∈(-F(E_n+1),F(-E_n+1)).
Let ⊂^n+1_+ be a compact, embedded, C^2-hypersurface with boundary ⊂_+^n+1 such that

  ν_F(x), -E_n+1=(x)≥_0,   for  any x∈∂Σ.

Then the anisotropic Gauss map ν_F:^F satisfies

{y∈^F:y, -E_n+1≤_0}
⊂ν_F(_+).



Proposition <ref> follows from  Proposition <ref> by introducing F̃ in (<ref>) and using Proposition <ref>.




We note that

{y∈^F:<y,-E_n+1>≤_0}+_0E_n+1^F
=^F_1(_0 E^F_n+1)∩_+^n+1,

and hence 
    ∫_{y∈^F:<y,-E_n+1>≤_0}F(ν(y)) A(y)=∫__1,_0^F∩^n+1_+F(ν(y)) A(y).

By virtue of  Proposition <ref>,
 we use the area formula in the following way,

   ∫__1,_0^F∩^n+1_+F(ν(y))A(y)
≤∫__+F(ν(p))H_n^F(p) A(p)

   ≤∫__+F(ν)(H^F)^n A
≤∫_F(ν)H^F^n A,

which is  (<ref>).

The proof of rigidity follows similarly as that in the proof of Theorems <ref>.




In view of (<ref>),

it is clear that the two Willmore-type inequalities (<ref>) and (<ref>) are different unless _0=0.






















































§ PROPERTIES OF UNBOUNDED CONVEX SETS


The following two Propositions may be familiar to experts, especially in the case  F(ξ)=ξ. 







Let K be an unbounded, closed convex set with boundary K

which contains the origin.
Fix 0≤ r<+∞,
then 
|^F_r+R∩ K|/R^n+1 is non-increasing. In particular, |^F_R∩ K|/R^n+1 is a constant if and only if K is a cone with vertex at the origin.



By using the co-area formula, for any 0≤ r, we have

/R|^F_r+R∩K|/R^n+1
=R^-(n+1)(∫_^F_r+R∩K1/|DF^o|A- (n+1)|^F_r+R∩K|/R).

On the other hand, by the divergence theorem, we have

(n+1)|^F_r+R∩K|=   ∫_^F_r+R∩Kdiv(x)x

=   ∫_^F_r+R∩Kx,DF^o(x)/DF^o(x)A
+∫_^F_r+R∩Kx,N̅(x)A

≥   ∫_^F_r+R∩KR/DF^oA.

In the last inequality, we have used x, N̅(x)≥ 0 thanks to the convexity of K.

It is then direct to see that

    / R|^F_r+R∩ K|/R^n+1≤ 0.


If
/ R|^F_R∩ K|/R^n+1= 0, then
x,N̅(x)=0 for ^n-a.e. x∈ K, it follows that K is a cone with vertex at the origin, see e.g., <cit.>*Proposition 28.8.


A direct consequence is that one can define the asymptotic volume ratio with respect to F for K as

 AVR_F(K)
 =lim_R→∞ |^F_R∩K|/^FR^n+1.




Similarly, we show the following asymptotic volume ratio for any F.


Let K be an unbounded, closed convex set with boundary K

which contains the origin.
There is a unique tangent cone at infinity of K, say K_∞.
Moreover, one has

AVR_F(K)
=|^F ∩K_∞|/^F.




Denote by K_R=1/RK. From the compactness result, see e.g., <cit.>*Corollary 12.27, we know that, up to a subsequence, say {R_h}_h∈,

K_R_hK_∞
for some sets of locally finite perimeter , with convergence in the sense that, for every compact set E⊂^n+1,

lim_h∞E∩(K_∞K_R_h)=0.

By the monotonicity, one can prove that K_∞ must be a cone. In fact,
for each s>0, we have

lim_h→∞|^F_s∩K_R_h|=lim_h→∞|^F_sR_h∩K|/R_h^n+1
(<ref>)=
s^n+1AVR_F(K)^F.

On the other hand, from (<ref>) we deduce

lim_h→∞|^F_s∩K_R_h|=|^F_s∩K_∞|.

It follows that, regardless of the choice of convergence subsequence, there always holds

|^F_s∩K_∞|/s^n+1
=AVR_F(K)^F.

As the LHS is a constant about s, it follows from Proposition <ref> that
 is a cone,
revealing the fact that

AVR_F(K)
=|^F∩K_∞|/^F.

In other words, we have proved that for any such non-compact convex set K, there exists a unique tangent cone at infinity, denoted by K_∞.
This completes the proof.





alpha


