# A Steiner Inequality for the Anisotropic Perimeter

Authors: Jin Dai
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

In this paper, we prove the monotonicity of the anisotropic perimeter of sets of finite perimeter under Steiner symmetrization by a variational formula of volume and an inequality for the anisotropic lower outer Minkowski content. As a consequence, we give a more direct proof of the Wulff inequality by Steiner symmetrization.


---

## Body

## 1. Introduction

LetE be a Lebesgue measurable set of ℝn,n≥2. The perimeter of E, PE, is defined as follows ([1, 2]):
(1)PE=sup∫EdivTxdx:T∈Cc1ℝn;ℝn,supℝnT≤1,where Cc1ℝn;ℝn denotes the class of all continuously differentiable ℝn-valued functions, with compact support in ℝn, and · is the Euclidean norm. If the perimeter PE of E is finite, that is, PE<∞, then we say that E is a set of finite perimeter.LetK⊂ℝn be a convex body (i.e., compact convex set with nonempty interior) containing the origin o in its interior. Let hK and ·K denote the support function and the gauge function of K, respectively (see Section 2 for the details). Given a set E of finite perimeter, the anisotropic perimeter (or surface energy) PKE of E with respect to K is given by ([3])
(2)PKE=sup∫EdivTxdx:T∈Cc1ℝn;ℝn,supℝnTK≤1.Here, the closed Euclidean unit ball ofℝn with center in the origin is denoted by B0,1. If K is equal to B0,1, then the anisotropic perimeter is the classical perimeter, that is, PB0,1E=PE. Generally, there exist constants c1K,c2K>0 only depending on K such that
(3)c2KPE≤PKE≤c1KPE.SoE has finite anisotropic perimeter if and only if it has finite Euclidean perimeter.Let us recall the statement of the celebrated Wulff inequality and also see [2, 4–11]. Here, the n-dimensional Lebesgue measure of a subset A of ℝn is denoted by A.Theorem 1.
LetK⊂ℝn be a convex body containing the origin in its interior, and let E⊂ℝn be a set of finite perimeter with E<∞. Then, we have
(4)PKE≥nK1/nEn−1/n.Equality holds in (4) if and only if E and K are homothetic up to a set of measure zero, i.e., ∃λ>0,x∈ℝn:EΔx+λK=0, where Δ denotes the operation of symmetric difference (see Section 2). Note that
(5)K=⋂u∈Sn−1x∈ℝn:x·u≤hKu,and K is known as the Wulff shape of hK, where Sn−1 denotes the unit sphere in ℝn and x·u denotes the Euclidean scalar product of x and u.The Wulff inequality is also known as Wulff problem, and the latter was first established by crystallographer Georg Wulff in [12]. It has been investigated intensively and plays an important role in many branches of geometry and analysis (for example, see [6, 13–16]). In particular, the famous isoperimetric inequality is a special case of the Wulff inequality, that is, setting K=B0,1 in (4), we see
(6)PB0,1E≥nB0,11/nEn−1/n,i.e.,(7)PE≥nωn1/nEn−1/n.Here,ωk=πk/2/Γ1+k/2 and kωk is the volume of the k-dimensional closed unit ball and the surface area of the k−1-dimensional unit sphere Sk−1 in ℝk, respectively, and Γ· is the standard gamma function. While there are multiple ways to establish the isoperimetric inequality, using Steiner symmetrization is especially elegant (e.g., see [17] and [18] for an English translation). In this paper, we will show that the Steiner symmetrization can be used to establish the Wulff inequality too. This approach has been used in [19–21] through an inequality of Klimov (see [20], Lemma 2, and also see [21], Lemma 5.12), while we will use this approach through the more geometric way. In particular, the notion of anisotropic lower outer Minkowski content was used and the related geometric inequalities were deduced (see Definition 2 and Section 3).Given a unit directionu∈Sn−1, the standard orthogonal basis of ℝn is denoted by e1,e2,⋯,en with u=en. We decompose ℝn as ℝn−1×ℝ, with the orthogonal projection p:ℝn⟶ℝn−1 and q:ℝn⟶ℝ, so that x=px,qx for x∈ℝn. For a Lebesgue measurable set E⊂ℝn, we define the vertical section Ez⊂ℝ of E as
(8)Ez=t∈ℝ:z,t∈E,z∈ℝn−1.The Steiner symmetrizationEu of E is then defined as
(9)Eu=x∈ℝn:qx≤12H1Epx,where Hk denotes the k-dimensional Hausdorff measure and Hk is equal to the Lebesgue measure in ℝk. By Fubini’s theorem, it is obvious that Lebesgue measure is invariant under Steiner symmetrization, that is, Eu=E. Steiner symmetrization has many excellent properties. A key property is Gross’s theorem, which states that iterating Steiner symmetrization of a convex body K through a suitable sequence of directions, the sequence of successive Steiner symmetrals of K converges to a Euclidean ball in the Hausdorff metric (see Section 4).Now, let us recall the known Steiner inequality, which says that the Euclidean perimeter ofE is decreasing under Steiner symmetrization, as follows.Lemma 1 ([2] pp.158) (Steiner inequality).
IfE is a set of finite perimeter in ℝn with E<∞ and a direction u∈Sn−1, then Eu is a set of finite perimeter in ℝn, with
(10)PEu≤PE.The Steiner inequality roughly ensures that the more symmetric set possesses the less perimeter. As a consequence, the famous isoperimetric inequality follows, and balls are unique extremums. Similarly, in this paper, we mainly consider the case of the anisotropic perimeter for the Steiner inequality. We prove the monotonicity of the anisotropic perimeter of sets of finite perimeter under Steiner symmetrization (see Theorem2) by a variation formula of volume (see Proposition 2) and an inequality for the anisotropic lower outer Minkowski content (see Lemma 7).Theorem 2.
LetK be a convex body containing the origin in its interior and let E be a set of finite perimeter with E<∞. For a unit direction u, we have
(11)PKuEu≤PKE.Note that ifK is a convex body containing the origin in its interior, the same holds also for Ku.We remark that this is a generalization of the Steiner inequality by replacingK with B0,1. In general, anisotropic perimeter does not behave monotonously under Steiner symmetrization (see [16]), that is, for a suitable convex body K there exist a direction u and a set E of finite perimeter such that
(12)PKEu>PKE.A function version of Theorem2 has been proved in [21] through an inequality of Klimov (see [21], Proposition 5.6).Now, we describe the structure in this paper. In Section2, we collect some elementary notations and necessary theorems. In Section 3, we prove Theorem 2 by using approximation by smooth sets. In Section 4, we conclude the proof of the Wulff inequality by using Theorem 2 (without the equality case).


## 2. Background Materials

In this section, some elementary materials are collected from convex geometry and geometric measure theory. For more information, see [2, 10, 22, 23].


### 2.1. Some Notations from Convex Geometry

The Minkowski sum of two subsetsA,B⊂ℝn is defined by
(13)A+B=a+b:a∈A,b∈B,and the scalar multiple of A by
(14)λA=λa:a∈A,for real numbers λ. We write −A for −1A, A−B for A+−B, and A+x for A+x, where x∈ℝn.A convex body is a compact convex subset ofℝn with nonempty interior. Each nonempty compact convex set K is uniquely determined by its support function hK, defined by
(15)hKx=supx·y:y∈K,for all x∈ℝn. Note that hK is positively homogeneous of degree 1 and subadditive. Conversely, each function with these two properties is the support function of a unique compact convex set. Let K be a convex body containing the origin o in its interior. The function ·K defined by
(16)xK=infλ≥0:x∈λK,for x∈ℝn is called the gauge function (or Minkowski functional) of K. Note that K is exactly equal to x∈ℝn:xK≤1 and the boundary ∂K of K is equal to x∈ℝn:xK=1. We write Kn for the space of convex bodies in ℝn endowed with the Hausdorff metric induced by the maximun norm of the support functions
(17)δHK,L=maxu∈Sn−1hKu−hLu.And we writeK0n for the class of convex bodies containing the origin in its interior. Given a sequence of convex bodies Kjj∈ℕ, we say that KjH-converge to a convex body K, and write Kj⟶HK, if
(18)limj⟶∞δHK,Kj=0.The following theorem combines the convergence of convex bodies with gauge functions of convex bodies.Theorem 3 ([24], pp.111).
LetKjj∈ℕ⊂Kon and K∈Kon. Then, the following statements are equivalent:
(1)
Kj⟶HK as j⟶∞(2)
supu∈Sn−1uK−uKj⟶0 as j⟶∞


### 2.2. Some Notations about Sets of Finite Perimeter

LetE and F be two Lebesgue measurable sets in ℝn. We define the symmetric difference metric of E and F by
(19)δVE,F=EΔF=E\F∪F\E.Given a sequence of Lebesgue measurable setsEjj∈ℕ, we say that Ej converges to E and write Ej⟶E, if
(20)limj⟶∞δVE,Ej=limj⟶∞EΔEj=0.Note thatδV and δH induce the same topology on Kn (see [23]), that is, Kj⟶HK is equivalent to Kj⟶K as j⟶∞.We use the notationCckℝn;ℝm with k=0,1,⋯,∞ to represent the class of all k-order continuously differentiable functions from ℝn to ℝm with compact support in ℝn. The following proposition gives an equivalent definition of sets of finite perimeter.Proposition 1 ([2] pp.122).
IfE is a Lebesgue measurable set in ℝn, then E is a set of finite perimeter if and only if there exists a ℝn-valued Radon measure μE on ℝn such that
(21)∫EdivT=∫ℝnT·dμE,∀T∈Cc1ℝn,ℝn,and the total variation μE is a finite Radon measure.We callμE the Gauss-Green measure of E and have PE=μEℝn.Note that ifE is a set of finite perimeter and EΔF=0, then F is a set of finite perimeter and μE=μF; the converse is also true. So, there is an equivalence relation in the class of sets of finite perimeter, that is, E~F if EΔF=0. Thus, we do not distinguish E and F if the symmetric difference of E and F is a set of Lebesgue measure zero.As mentioned above, we may modify a set of finite perimeterE by a set of measure zero without changing its Gauss-Green measure. So, moreover, we can give a notion of reduced boundary. We denote the ball with center x∈ℝn and radius r>0 by Bx,r.Definition 1 ([2], pp.165).
The reduced boundary∂∗E of a set of finite perimeter E in ℝn is the set of those x∈sptμE (the support of μE) such that the limit
(22)limr⟶0+μEBx,rμEBx,rexistsandbelongstoSn−1.
We define a Borel functionνE:∂∗E⟶Sn−1 by setting
(23)νEx=limr⟶0+μEBx,rμEBx,rforx∈∂∗E.
We callνE the measure-theoretic outer unit normal to E.Note that in factνE is the Radon-Nikodym derivative of μE with respect to μE. Next, we state De Giorgi’s structure theorem, which states that the reduced boundary has the structure of a generalized hypersurfaces.Theorem 4 ([2] pp.170).
IfE is a set of finite perimeter in ℝn, then the Gauss-Green measure μE of E satisfies
(24)μE=νEHn−1⌞∂∗E,μE=Hn−1⌞∂∗E,and the generalized Gauss-Green formula holds true:
(25)∫EdivT=∫∂∗ET·νEdHn−1,∀T∈Cc1ℝn,ℝn.From the structure of sets of finite perimeter and some computations, the anisotropic perimeter ofE can be represented as (see Theorem 6)
(26)PKE=∫∂∗EhKνExdHn−1x.Using (2), (25), and the denseness of Cc∞ℝn;ℝn in Cc1ℝn;ℝn, we also have
(27)PKE=sup∫EdivTxdx:T∈Cc∞ℝn;ℝn,supℝnTK≤1=sup∫∂∗ET·νEdHn−1:T∈Cc∞ℝn;ℝn,supℝnTK≤1.We now give a notion of measure for the boundary of a body or domain inℝn.Definition 2.
LetE be a bounded Lebesgue measurable set in ℝn, K∈Kon and t>0. Setting
(28)ΛE,K,t=E+tK−Et,we define the lower outer Minkowski content SMK∗E of E with respect to K as
(29)SMK∗E=liminft⟶0+ΛE,K,t.
Simply, we callSMK∗E the anisotropic lower outer Minkowski content.Anisotropic lower outer Minkowski content is deeply related to anisotropic perimeter and the following lemma is useful.Lemma 2 ([25]).
LetM be a compact domain with piecewise C1 boundary ∂M, and K∈Kon. Then,
(30)∫∂MhKνMxdHn−1x=SMK∗M.We remark that the equality (30) generally does not hold true for a set of finite perimeter, but a weak result will be proved in Section 3 (see Lemma 8).Next, we list the fundamental approximation theorem and representation theorem of anisotropic perimeter.Definition 3 ([2] pp.41).
Letμjj∈ℕ and μ be Radon measures on ℝn with values in ℝm, m∈ℕ. We say that μj weak-star converges to μ and write μj⇀∗μ, if
(31)∫ℝnϕ·dμ=limj⟶∞∫ℝnϕ·dμj,∀ϕ∈Cc0ℝn,ℝm.
A setE⊂ℝn is called smooth if its boundary is a C∞ submanifold of ℝn.Theorem 5 ([2] pp.150) (approximation by smooth sets).
IfE is a set of finite perimeter in ℝn, with E<∞, then there exists a sequence Ejj∈ℕ of bounded open sets with smooth boundary in ℝn, such that
(32)Ej⟶E,PEj⟶PE,μEj⇀∗μE.Notice that the topological boundary of a smooth bounded setE is a set of Lebesgue measure zero, that is, ∂E=0. Hence, E is equivalent to the interior intE of E and the closure clE of E, respectively. Therefore, the sequence of smooth bounded open sets above can be replaced by a sequence of smooth compact sets when approximating a set of finite perimeter.In [2], the author defines anisotropic perimeter as
(33)∫∂∗EhK∘νEdHn−1,while anisotropic perimeter is defined as (2) in this paper. In fact, the two definitions of anisotropic perimeter do coincide as pointed out by Theorem 6.Theorem 6 ([26]) (representation of anisotropic perimeter).
IfE is a set of finite perimeter and K∈Kon, then we have
(34)PKE=∫∂∗EhK∘νEdHn−1.Finally, we gather some necessary properties of sets of finite perimeter for proving Theorem2.Lemma 3 ([2], pp.126) (lower semicontinuity of perimeter).
IfE is a Lebesgue measurable set and Ejj∈ℕ is a sequence of sets of finite perimeter in ℝn, with
(35)Ej⟶E,limsupj⟶∞PEj<∞,then E is of finite perimeter in ℝn, μEj⇀∗μE and
(36)PE≤liminfj⟶∞PEj.Lemma 4 ([2] pp.259) (lower semicontinuity of anisotropic perimeter).
GivenK∈Kon, if Ejj∈ℕ and E are sets of finite perimeter with Ej⟶E and μEj⇀∗μE, then
(37)PKE≤liminfj⟶∞PKEj.Lemma 5 ([2], pp.261) (continuity of anisotropic perimeter).
GivenK∈Kon, if Ejj∈ℕ and E are sets of finite perimeter with Ej⟶E, μEj⇀∗μE and PEj⟶PE, then
(38)PKE=limj⟶∞PKEj.


## 2.1. Some Notations from Convex Geometry

The Minkowski sum of two subsetsA,B⊂ℝn is defined by
(13)A+B=a+b:a∈A,b∈B,and the scalar multiple of A by
(14)λA=λa:a∈A,for real numbers λ. We write −A for −1A, A−B for A+−B, and A+x for A+x, where x∈ℝn.A convex body is a compact convex subset ofℝn with nonempty interior. Each nonempty compact convex set K is uniquely determined by its support function hK, defined by
(15)hKx=supx·y:y∈K,for all x∈ℝn. Note that hK is positively homogeneous of degree 1 and subadditive. Conversely, each function with these two properties is the support function of a unique compact convex set. Let K be a convex body containing the origin o in its interior. The function ·K defined by
(16)xK=infλ≥0:x∈λK,for x∈ℝn is called the gauge function (or Minkowski functional) of K. Note that K is exactly equal to x∈ℝn:xK≤1 and the boundary ∂K of K is equal to x∈ℝn:xK=1. We write Kn for the space of convex bodies in ℝn endowed with the Hausdorff metric induced by the maximun norm of the support functions
(17)δHK,L=maxu∈Sn−1hKu−hLu.And we writeK0n for the class of convex bodies containing the origin in its interior. Given a sequence of convex bodies Kjj∈ℕ, we say that KjH-converge to a convex body K, and write Kj⟶HK, if
(18)limj⟶∞δHK,Kj=0.The following theorem combines the convergence of convex bodies with gauge functions of convex bodies.Theorem 3 ([24], pp.111).
LetKjj∈ℕ⊂Kon and K∈Kon. Then, the following statements are equivalent:
(1)
Kj⟶HK as j⟶∞(2)
supu∈Sn−1uK−uKj⟶0 as j⟶∞


## 2.2. Some Notations about Sets of Finite Perimeter

LetE and F be two Lebesgue measurable sets in ℝn. We define the symmetric difference metric of E and F by
(19)δVE,F=EΔF=E\F∪F\E.Given a sequence of Lebesgue measurable setsEjj∈ℕ, we say that Ej converges to E and write Ej⟶E, if
(20)limj⟶∞δVE,Ej=limj⟶∞EΔEj=0.Note thatδV and δH induce the same topology on Kn (see [23]), that is, Kj⟶HK is equivalent to Kj⟶K as j⟶∞.We use the notationCckℝn;ℝm with k=0,1,⋯,∞ to represent the class of all k-order continuously differentiable functions from ℝn to ℝm with compact support in ℝn. The following proposition gives an equivalent definition of sets of finite perimeter.Proposition 1 ([2] pp.122).
IfE is a Lebesgue measurable set in ℝn, then E is a set of finite perimeter if and only if there exists a ℝn-valued Radon measure μE on ℝn such that
(21)∫EdivT=∫ℝnT·dμE,∀T∈Cc1ℝn,ℝn,and the total variation μE is a finite Radon measure.We callμE the Gauss-Green measure of E and have PE=μEℝn.Note that ifE is a set of finite perimeter and EΔF=0, then F is a set of finite perimeter and μE=μF; the converse is also true. So, there is an equivalence relation in the class of sets of finite perimeter, that is, E~F if EΔF=0. Thus, we do not distinguish E and F if the symmetric difference of E and F is a set of Lebesgue measure zero.As mentioned above, we may modify a set of finite perimeterE by a set of measure zero without changing its Gauss-Green measure. So, moreover, we can give a notion of reduced boundary. We denote the ball with center x∈ℝn and radius r>0 by Bx,r.Definition 1 ([2], pp.165).
The reduced boundary∂∗E of a set of finite perimeter E in ℝn is the set of those x∈sptμE (the support of μE) such that the limit
(22)limr⟶0+μEBx,rμEBx,rexistsandbelongstoSn−1.
We define a Borel functionνE:∂∗E⟶Sn−1 by setting
(23)νEx=limr⟶0+μEBx,rμEBx,rforx∈∂∗E.
We callνE the measure-theoretic outer unit normal to E.Note that in factνE is the Radon-Nikodym derivative of μE with respect to μE. Next, we state De Giorgi’s structure theorem, which states that the reduced boundary has the structure of a generalized hypersurfaces.Theorem 4 ([2] pp.170).
IfE is a set of finite perimeter in ℝn, then the Gauss-Green measure μE of E satisfies
(24)μE=νEHn−1⌞∂∗E,μE=Hn−1⌞∂∗E,and the generalized Gauss-Green formula holds true:
(25)∫EdivT=∫∂∗ET·νEdHn−1,∀T∈Cc1ℝn,ℝn.From the structure of sets of finite perimeter and some computations, the anisotropic perimeter ofE can be represented as (see Theorem 6)
(26)PKE=∫∂∗EhKνExdHn−1x.Using (2), (25), and the denseness of Cc∞ℝn;ℝn in Cc1ℝn;ℝn, we also have
(27)PKE=sup∫EdivTxdx:T∈Cc∞ℝn;ℝn,supℝnTK≤1=sup∫∂∗ET·νEdHn−1:T∈Cc∞ℝn;ℝn,supℝnTK≤1.We now give a notion of measure for the boundary of a body or domain inℝn.Definition 2.
LetE be a bounded Lebesgue measurable set in ℝn, K∈Kon and t>0. Setting
(28)ΛE,K,t=E+tK−Et,we define the lower outer Minkowski content SMK∗E of E with respect to K as
(29)SMK∗E=liminft⟶0+ΛE,K,t.
Simply, we callSMK∗E the anisotropic lower outer Minkowski content.Anisotropic lower outer Minkowski content is deeply related to anisotropic perimeter and the following lemma is useful.Lemma 2 ([25]).
LetM be a compact domain with piecewise C1 boundary ∂M, and K∈Kon. Then,
(30)∫∂MhKνMxdHn−1x=SMK∗M.We remark that the equality (30) generally does not hold true for a set of finite perimeter, but a weak result will be proved in Section 3 (see Lemma 8).Next, we list the fundamental approximation theorem and representation theorem of anisotropic perimeter.Definition 3 ([2] pp.41).
Letμjj∈ℕ and μ be Radon measures on ℝn with values in ℝm, m∈ℕ. We say that μj weak-star converges to μ and write μj⇀∗μ, if
(31)∫ℝnϕ·dμ=limj⟶∞∫ℝnϕ·dμj,∀ϕ∈Cc0ℝn,ℝm.
A setE⊂ℝn is called smooth if its boundary is a C∞ submanifold of ℝn.Theorem 5 ([2] pp.150) (approximation by smooth sets).
IfE is a set of finite perimeter in ℝn, with E<∞, then there exists a sequence Ejj∈ℕ of bounded open sets with smooth boundary in ℝn, such that
(32)Ej⟶E,PEj⟶PE,μEj⇀∗μE.Notice that the topological boundary of a smooth bounded setE is a set of Lebesgue measure zero, that is, ∂E=0. Hence, E is equivalent to the interior intE of E and the closure clE of E, respectively. Therefore, the sequence of smooth bounded open sets above can be replaced by a sequence of smooth compact sets when approximating a set of finite perimeter.In [2], the author defines anisotropic perimeter as
(33)∫∂∗EhK∘νEdHn−1,while anisotropic perimeter is defined as (2) in this paper. In fact, the two definitions of anisotropic perimeter do coincide as pointed out by Theorem 6.Theorem 6 ([26]) (representation of anisotropic perimeter).
IfE is a set of finite perimeter and K∈Kon, then we have
(34)PKE=∫∂∗EhK∘νEdHn−1.Finally, we gather some necessary properties of sets of finite perimeter for proving Theorem2.Lemma 3 ([2], pp.126) (lower semicontinuity of perimeter).
IfE is a Lebesgue measurable set and Ejj∈ℕ is a sequence of sets of finite perimeter in ℝn, with
(35)Ej⟶E,limsupj⟶∞PEj<∞,then E is of finite perimeter in ℝn, μEj⇀∗μE and
(36)PE≤liminfj⟶∞PEj.Lemma 4 ([2] pp.259) (lower semicontinuity of anisotropic perimeter).
GivenK∈Kon, if Ejj∈ℕ and E are sets of finite perimeter with Ej⟶E and μEj⇀∗μE, then
(37)PKE≤liminfj⟶∞PKEj.Lemma 5 ([2], pp.261) (continuity of anisotropic perimeter).
GivenK∈Kon, if Ejj∈ℕ and E are sets of finite perimeter with Ej⟶E, μEj⇀∗μE and PEj⟶PE, then
(38)PKE=limj⟶∞PKEj.


## 3. The Monotonicity of Anisotropic Perimeter under Steiner Symmetrization

In this section, the main result of the paper, namely, Theorem2, will be proved. Firstly, we state some elementary properties for Steiner symmetrization.Lemma 6.
Given a unit directionu and two compact sets E, F in ℝn, Steiner symmetrization of E and F through the direction u has the following properties:
(i)
λEu=λEu, for λ>0(ii)
E+Fu⊃Eu+Fu(iii)
EuΔFu≤EΔFProof.
Firstly, we provei. Note that λEpx=λEpx/λ for each λ>0 and x∈ℝn. We directly calculate
(39)λEu=x∈ℝn:qx≤12H1λEpx=λxλ:qx≤λ12H1Epx/λ=λxλ:qxλ≤12H1Epx/λ=λy∈ℝn:qy≤12H1Epy=λEu.
We now proveii. The linear subspace spanu spanned by u is denoted as L. Let x∈Eu, y∈Fu or, equivalently, x=px,qx, y=py,qy, where
(40)qx≤12H1Epx=12H1E∩L+x,qy≤12H1Fpy=12H1F∩L+y.
Then, we have(41)qx+y≤qx+qy≤12H1E∩L+x+H1F∩L+y≤12H1E∩L+x+F∩L+y≤12H1E+F∩L+x+y=12H1E+Fpx+y.
Here, (41) holds by applying the 1-dimensional Brunn-Minkowski inequality for compact sets (see [10], pp.146) and E∩L+x+F∩L+y⊂E+F∩L+x+y. Hence, x+y∈E+Fu.
Finally, we complete the proof ofiii. Since, Steiner symmetrization preserves inclusion relation, we find
(42)E∩Fu⊂Eu∩Fu.
Then,(43)Eu\Fu=Eu−Eu∩Fu≤Eu−E∩Fu=E−E∩F=E\F.
Similarly, we obtainFu\Eu≤F\E, so
(44)EuΔFu≤EΔF.The following lemma shows that the anisotropic lower outer Minkowski content decreases under Steiner symmetrization.Lemma 7.
IfE is a compact set, K∈Kon and u∈Sn−1, then we have
(45)SMKu∗Eu≤SMK∗E.Proof.
Byi and ii of Lemma 6,
(46)E+tKu⊃Eu+tKufort>0.
UsingEu=E, then
(47)Eu+tKu−Eut≤E+tKu−Eut=E+tK−Etfort>0.
Now, lett⟶0+, so we get (45).Next, we state a formula of first variation of volume (see [24]). A one-parameter family of diffeomorphisms of ℝn is a smooth function
(48)x,t∈ℝn×−ε,ε↦ft,x=ftx∈ℝn,ε>0,such that, for each fixed t<ε, ft:ℝn⟶ℝn is a diffeomorphism of ℝn. Given an open set A in ℝn, we say that ftt<ε is a local variation in A if it defines a one-parameter family of diffeomorphisms such that
(49)f0x=x,∀x∈ℝn,x∈ℝn:ftx≠x⊂⊂A,∀t<ε.Here,ot means that ot/t⟶0 as t⟶0, and o1 means that o1⟶0 as t⟶0. By Taylor’s expansion, the following holds uniformly on ℝn:
(50)ftx=x+tTx+ot,∇ftx=Id+t∇Tx+ot,where T∈Cc∞A;ℝn is the initial velocity of ftt<ε,
(51)Tx=∂ft∂tx,0,x∈ℝn.Conversely, starting fromT∈Cc∞A;ℝn, we can set ftx=x+tTx,x∈ℝn to construct a local variation ftt<ε in A having T as its initial velocity.Proposition 2 ([2] pp.202) (first variation of volume).
IfE is a set of finite perimeter in ℝn, E<∞, A is open, and ftt<ε is a local variation in A with initial velocity T, then
(52)ftE=E+t∫∂∗ET·νEdHn−1+ot.Now, we claim that the anisotropic perimeter ofE is less than or equal to the anisotropic outer Minkowski content of E by application of first variation of volume.Lemma 8.
GivenK∈Kon, if E is a bounded set of finite perimeter, then
(53)PKE≤SMK∗E.Proof.
GivenT∈Cc∞ℝn,ℝn with supx∈ℝnTxK≤1, we set ftx=x+tTx,x∈ℝn, and we construct a variation ftt<ε in ℝn. By Proposition 2, we have
(54)ftE=E+t∫∂∗ET·νEdHn−1+ot.
That is(55)ftE−Et=∫∂∗ET·νEdHn−1+o1.
On the other hand, for eachx+tTx∈ftE, x∈E, since supx∈ℝnTxK≤1, then
(56)Tx∈K=y∈ℝn:yK≤1.
Thus,x+tTx∈E+tK, that is ftE⊂E+tK. Then, we have
(57)∫∂∗ET·νEdHn−1+o1=ftE−Et≤E+tK−Et.
Lett⟶0+, so we get
(58)∫∂∗ET·νEdHn−1≤SMK∗E,for all T∈Cc∞ℝn;ℝn with supx∈ℝnTxK≤1. Taking supremum for Tin the inequality (58), we get
(59)PKE≤SMK∗E.Now, we prove Theorem2.Proof.
GivenK∈Kon, let E be a compact set with smooth boundary. By Theorem 6, Lemma 2, and Lemma 7, we have
(60)PKE=∫∂EhKνExdHn−1x=SMK∗E≥SMKu∗Eu.
SinceEu is of finite perimeter (see Lemma 1) and compact, we obtain
(61)PKE≥SMKu∗Eu≥PKuEu,by applying Lemma 8. Hence, we complete the proof of Theorem 2 for each smooth compact set.
Now, using Theorem5, we select a sequence of compact sets Ejj∈ℕ with smooth boundary to approximate the set of finite perimeter E, that is
(62)Ej⟶E,PEj⟶PE,μEj⇀∗μE,(63)PKuEju≤PKEj,where we write Eju for Eju. By iii of Lemma 6, we find EjuΔEu≤EjΔE, which implies that Eju⟶Eu. We also have
(64)limsupj⟶∞PEju≤limsupj⟶∞PEj<PE+1.
Hence, according to Lemma3, we obtain μEju⇀∗μEu. Therefore, it follows from Lemma 4 that
(65)PKuEu≤liminfj⟶∞PKuEju.
On the other hand, thanks to (62) and Lemma 5, we get
(66)PKE=limj⟶∞PKEj.
Lettingj⟶∞ in (63), we obtain that
(67)PKuEu≤PKE.


## 4. Proofs of the Wulff Inequality

In this section, we prove the Wulff inequality by applying Theorem2. It is a Steiner symmetrization approach to this inequality. First, we give some interesting lemmas.The following is a semicontinuity property of anisotropic perimeter.Lemma 9.
IfKjj∈ℕ⊂Kon, K∈Kon with Kj⟶HK, and Ejj∈ℕ, E are sets of finite perimeter with Ej⟶E and supj∈ℕPEj<∞, then we have
(68)PKE≤liminfj⟶∞PKjEj.Proof.
Using Lemma4, we have
(69)PKE≤liminfj⟶∞PKEj.
We directly calculate(70)PKjEj−PKEj=∫∂∗EjhKjνEjx−hKνEjxdHn−1x≤∫∂∗EjhKjνEjx−hKνEjxdHn−1x≤maxu∈Sn−1hKju−hKuPEj≤δHKj,Ksupj∈ℕPEj.
Then, we get(71)liminfj⟶∞PKjEj−PKEj=0.
Therefore, we have(72)liminfj⟶∞PKjEj=liminfj⟶∞PKjEj−PKEj+PKEj≥liminfj⟶∞PKjEj−PKEj+liminfj⟶∞PKEj=liminfj⟶∞PKEj≥PKE.The anisotropic perimeter with respect toK is positively homogeneous of degree 1 for K.Lemma 10.
Given a setE of finite perimeter, K∈Kon, and a real number λ>0, then
(73)PλKE=λPKE.Proof.
We directly compute(74)PλKE=∫∂∗EhλKνExdHn−1x=∫∂∗EλhKνExdHn−1x=λ∫∂∗EhKνExdHn−1x=λPKE,by the positive homogeneity of degree 1 of the support function hK.Givenujj∈ℕ⊂Sn−1 and a set of finite perimeter E, we denote by Ei the following object:
(75)Ei≔⋯Eu1u2⋯ui.The intuitive sphericity theorem of Gross is important for our proof, which reads as follows.Lemma 11 ([10] pp.172).
LetK∈Kn. Then, there exists a sequence ujj∈ℕ of unit directions such that
(76)limj⟶∞Kj=Kωn1/nB0,1,with respect to the Hausdorff metric.It shows that by suitably doing repeated Steiner symmetrizations of a convex bodyK one obtains a sequence of convex bodies which converge to a Euclidean ball having the same volume as K.Finally, we also need a compactness criterion for sets of finite perimeter.Lemma 12 ([2], pp.132) (compactness).
IfR>0 and Ejj∈ℕ are sets of finite perimeter in ℝn, with
(77)supj∈ℕPEj<∞,Ej⊂B0,R,∀j∈ℕ,then there exists E of finite perimeter in ℝn and jk⟶∞ as k⟶∞, with
(78)Ejk⟶E,μEjk⇀∗μE,E⊂B0,R.In the following, we prove Theorem1.Proof.
LetE be a compact set with smooth boundary and K∈Kon. By Lemma 11, there exists a sequence ujj∈ℕ of unit directions such that
(79)Kj⟶HKωn1/nB0,1,as j⟶∞. We successively apply Steiner symmetrization through the sequence ujj∈ℕ for initial bodies K and E. By application of Theorem 2, we have a chain of inequalities
(80)PKE≥PK1E1≥PK2E2≥⋯≥PKiEi≥⋯.
For eachi∈ℕ, it follows from the Steiner inequality that PEi≤PE, so
(81)supj∈ℕPEj≤PE<∞.
SinceE is compact, there exists a real number R>0 such that E⊂B0,R and Ei⊂B0,R for all i∈ℕ. Hence, in addition to (81), we can use Lemma 12. Hence, there exists E0 of finite perimeter in ℝn and jk⟶∞ as k⟶∞, with
(82)Ejk⟶E0,μEjk⇀∗μE0,while
(83)Kjk⟶HKωn1/nB0,1.
Since (81), (82), and (83) hold, we can apply Lemma 9. Therefore,
(84)PKE≥liminfk⟶∞PKjkEjk≥PK/ωn1/nB0,1E0.
So we can calculate(85)PKE≥PK/ωn1/nB0,1E0=Kωn1/nPB0,1E0=Kωn1/nPE0≥Kωn1/nnωn1/nE0n−1/n=nK1/nE0n−1/n,where Lemma 10 and the Euclidean isoperimetric inequality (7) were used. Note that
(86)Ej⟶FasEj⟶F,where Ej, j∈ℕ and F are Lebesgue measurable sets. Then, we obtain
(87)E0=limk⟶∞Ejk=limk⟶∞E=E.
Hence, we proved the Wulff inequality for each smooth compact set.
Next, according to Theorem5, we select a sequence Ejj∈ℕ of smooth compact sets to approximate a set E of finite perimeter, and Ejj∈ℕsatisfy
(88)PKEj≥nK1/nEjn−1/n,∀j∈ℕ.
Letj⟶∞, and then, thanks to Lemma 5 and (86), so we get
(89)PKE≥nK1/nEn−1/n.



---

*Source: 2989121-2022-04-30.xml*