A Steiner Inequality for the Anisotropic Perimeter

Dai, Jin

doi:https://doi.org/10.1155/2022/2989121

Journal of Function Spaces

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Research Article | Open Access

Volume 2022 | Article ID 2989121 | https://doi.org/10.1155/2022/2989121

A Steiner Inequality for the Anisotropic Perimeter

Jin Dai¹

Academic Editor: Serena Matucci

Received17 Feb 2022

Accepted05 Apr 2022

Published30 Apr 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.

1. Introduction

Let be a Lebesgue measurable set of . The perimeter of , , is defined as follows ([1, 2]): where denotes the class of all continuously differentiable -valued functions, with compact support in , and is the Euclidean norm. If the perimeter of is finite, that is, , then we say that is a set of finite perimeter.

Let be a convex body (i.e., compact convex set with nonempty interior) containing the origin in its interior. Let and denote the support function and the gauge function of , respectively (see Section 2 for the details). Given a set of finite perimeter, the anisotropic perimeter (or surface energy) of with respect to is given by ([3])

Here, the closed Euclidean unit ball of with center in the origin is denoted by . If is equal to , then the anisotropic perimeter is the classical perimeter, that is, . Generally, there exist constants only depending on such that

So 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 -dimensional Lebesgue measure of a subset of is denoted by .

Theorem 1. Let be a convex body containing the origin in its interior, and let be a set of finite perimeter with . Then, we have

Equality holds in (4) if and only if and are homothetic up to a set of measure zero, i.e., , where denotes the operation of symmetric difference (see Section 2). Note that and is known as the Wulff shape of , where denotes the unit sphere in and denotes the Euclidean scalar product of and .

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 in (4), we see

i.e.,

Here, and is the volume of the -dimensional closed unit ball and the surface area of the -dimensional unit sphere in , respectively, and 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 direction , the standard orthogonal basis of is denoted by with . We decompose as , with the orthogonal projection and , so that for . For a Lebesgue measurable set , we define the vertical section of as

The Steiner symmetrization of is then defined as where denotes the -dimensional Hausdorff measure and is equal to the Lebesgue measure in . By Fubini’s theorem, it is obvious that Lebesgue measure is invariant under Steiner symmetrization, that is, . Steiner symmetrization has many excellent properties. A key property is Gross’s theorem, which states that iterating Steiner symmetrization of a convex body through a suitable sequence of directions, the sequence of successive Steiner symmetrals of 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 of is decreasing under Steiner symmetrization, as follows.

Lemma 1 ([2] pp.158) (Steiner inequality). If is a set of finite perimeter in with and a direction , then is a set of finite perimeter in , with

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 Theorem 2) by a variation formula of volume (see Proposition 2) and an inequality for the anisotropic lower outer Minkowski content (see Lemma 7).

Theorem 2. Let be a convex body containing the origin in its interior and let be a set of finite perimeter with . For a unit direction , we have

Note that if is a convex body containing the origin in its interior, the same holds also for .

We remark that this is a generalization of the Steiner inequality by replacing with . In general, anisotropic perimeter does not behave monotonously under Steiner symmetrization (see [16]), that is, for a suitable convex body there exist a direction and a set of finite perimeter such that

A function version of Theorem 2 has been proved in [21] through an inequality of Klimov (see [21], Proposition 5.6).

Now, we describe the structure in this paper. In Section 2, 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 subsets is defined by and the scalar multiple of by for real numbers . We write for , for , and for , where .

A convex body is a compact convex subset of with nonempty interior. Each nonempty compact convex set is uniquely determined by its support function , defined by for all . Note that 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 be a convex body containing the origin in its interior. The function defined by for is called the gauge function (or Minkowski functional) of . Note that is exactly equal to and the boundary of is equal to . We write for the space of convex bodies in endowed with the Hausdorff metric induced by the maximun norm of the support functions

And we write for the class of convex bodies containing the origin in its interior. Given a sequence of convex bodies , we say that -converge to a convex body , and write , if

The following theorem combines the convergence of convex bodies with gauge functions of convex bodies.

Theorem 3 ([24], pp.111). Let and . Then, the following statements are equivalent: (1) as (2) as

2.2. Some Notations about Sets of Finite Perimeter

Let and be two Lebesgue measurable sets in . We define the symmetric difference metric of and by

Given a sequence of Lebesgue measurable sets , we say that converges to and write , if

Note that and induce the same topology on (see [23]), that is, is equivalent to as .

We use the notation with to represent the class of all -order continuously differentiable functions from to with compact support in . The following proposition gives an equivalent definition of sets of finite perimeter.

Proposition 1 ([2] pp.122). If is a Lebesgue measurable set in , then is a set of finite perimeter if and only if there exists a -valued Radon measure on such that and the total variation is a finite Radon measure.

We call the Gauss-Green measure of and have .

Note that if is a set of finite perimeter and , then is a set of finite perimeter and ; the converse is also true. So, there is an equivalence relation in the class of sets of finite perimeter, that is, if . Thus, we do not distinguish and if the symmetric difference of and is a set of Lebesgue measure zero.

As mentioned above, we may modify a set of finite perimeter 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 and radius by .

Definition 1 ([2], pp.165). The reduced boundary of a set of finite perimeter in is the set of those (the support of ) such that the limit We define a Borel function by setting We call the measure-theoretic outer unit normal to .

Note that in fact is the Radon-Nikodym derivative of with respect to . 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). If is a set of finite perimeter in , then the Gauss-Green measure of satisfies and the generalized Gauss-Green formula holds true:

From the structure of sets of finite perimeter and some computations, the anisotropic perimeter of can be represented as (see Theorem 6)

Using (2), (25), and the denseness of in , we also have

We now give a notion of measure for the boundary of a body or domain in .

Definition 2. Let be a bounded Lebesgue measurable set in , and . Setting we define the lower outer Minkowski content of with respect to as Simply, we call 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]). Let be a compact domain with piecewise boundary , and . Then,

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). Let and be Radon measures on with values in , . We say that weak-star converges to and write , if A set is called smooth if its boundary is a submanifold of .

Theorem 5 ([2] pp.150) (approximation by smooth sets). If is a set of finite perimeter in , with , then there exists a sequence of bounded open sets with smooth boundary in , such that

Notice that the topological boundary of a smooth bounded set is a set of Lebesgue measure zero, that is, . Hence, is equivalent to the interior of and the closure of , 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 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). If is a set of finite perimeter and , then we have

Finally, we gather some necessary properties of sets of finite perimeter for proving Theorem 2.

Lemma 3 ([2], pp.126) (lower semicontinuity of perimeter). If is a Lebesgue measurable set and is a sequence of sets of finite perimeter in , with then is of finite perimeter in , and

Lemma 4 ([2] pp.259) (lower semicontinuity of anisotropic perimeter). Given , if and are sets of finite perimeter with and , then

Lemma 5 ([2], pp.261) (continuity of anisotropic perimeter). Given , if and are sets of finite perimeter with , and , then

3. The Monotonicity of Anisotropic Perimeter under Steiner Symmetrization

In this section, the main result of the paper, namely, Theorem 2, will be proved. Firstly, we state some elementary properties for Steiner symmetrization.

Lemma 6. Given a unit direction and two compact sets , in , Steiner symmetrization of and through the direction has the following properties: (i), for (ii)(iii)

Proof. Firstly, we prove . Note that for each and . We directly calculate We now prove . The linear subspace spanned by is denoted as . Let , or, equivalently, , , where Then, we have Here, (41) holds by applying the -dimensional Brunn-Minkowski inequality for compact sets (see [10], pp.146) and . Hence, .
Finally, we complete the proof of . Since, Steiner symmetrization preserves inclusion relation, we find Then, Similarly, we obtain , so

The following lemma shows that the anisotropic lower outer Minkowski content decreases under Steiner symmetrization.

Lemma 7. If is a compact set, and , then we have

Proof. By and of Lemma 6, Using , then Now, let , so we get (45).

Next, we state a formula of first variation of volume (see [24]). A one-parameter family of diffeomorphisms of is a smooth function such that, for each fixed , is a diffeomorphism of . Given an open set in , we say that is a local variation in if it defines a one-parameter family of diffeomorphisms such that

Here, means that as , and means that as . By Taylor’s expansion, the following holds uniformly on : where is the initial velocity of ,

Conversely, starting from , we can set to construct a local variation in having as its initial velocity.

Proposition 2 ([2] pp.202) (first variation of volume). If is a set of finite perimeter in , , is open, and is a local variation in with initial velocity , then

Now, we claim that the anisotropic perimeter of is less than or equal to the anisotropic outer Minkowski content of by application of first variation of volume.

Lemma 8. Given , if is a bounded set of finite perimeter, then

Proof. Given with , we set , and we construct a variation in . By Proposition 2, we have That is On the other hand, for each , , since , then Thus, , that is . Then, we have Let , so we get for all with Taking supremum for in the inequality (58), we get

Now, we prove Theorem 2.

Proof. Given , let be a compact set with smooth boundary. By Theorem 6, Lemma 2, and Lemma 7, we have Since is of finite perimeter (see Lemma 1) and compact, we obtain by applying Lemma 8. Hence, we complete the proof of Theorem 2 for each smooth compact set.
Now, using Theorem 5, we select a sequence of compact sets with smooth boundary to approximate the set of finite perimeter , that is where we write for . By of Lemma 6, we find , which implies that . We also have Hence, according to Lemma 3, we obtain . Therefore, it follows from Lemma 4 that On the other hand, thanks to (62) and Lemma 5, we get Letting in (63), we obtain that

4. Proofs of the Wulff Inequality

In this section, we prove the Wulff inequality by applying Theorem 2. 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. If , with , and , are sets of finite perimeter with and , then we have

Proof. Using Lemma 4, we have We directly calculate Then, we get Therefore, we have

The anisotropic perimeter with respect to is positively homogeneous of degree 1 for .

Lemma 10. Given a set of finite perimeter, , and a real number , then

Proof. We directly compute by the positive homogeneity of degree 1 of the support function .

Given and a set of finite perimeter , we denote by the following object:

The intuitive sphericity theorem of Gross is important for our proof, which reads as follows.

Lemma 11 ([10] pp.172). Let . Then, there exists a sequence of unit directions such that with respect to the Hausdorff metric.

It shows that by suitably doing repeated Steiner symmetrizations of a convex body one obtains a sequence of convex bodies which converge to a Euclidean ball having the same volume as .

Finally, we also need a compactness criterion for sets of finite perimeter.

Lemma 12 ([2], pp.132) (compactness). If and are sets of finite perimeter in , with then there exists of finite perimeter in and as , with

In the following, we prove Theorem 1.

Proof. Let be a compact set with smooth boundary and . By Lemma 11, there exists a sequence of unit directions such that as . We successively apply Steiner symmetrization through the sequence for initial bodies and . By application of Theorem 2, we have a chain of inequalities For each , it follows from the Steiner inequality that , so Since is compact, there exists a real number such that and for all . Hence, in addition to (81), we can use Lemma 12. Hence, there exists of finite perimeter in and as , with while Since (81), (82), and (83) hold, we can apply Lemma 9. Therefore, So we can calculate where Lemma 10 and the Euclidean isoperimetric inequality (7) were used. Note that where , and are Lebesgue measurable sets. Then, we obtain Hence, we proved the Wulff inequality for each smooth compact set.
Next, according to Theorem 5, we select a sequence of smooth compact sets to approximate a set of finite perimeter, and satisfy Let , and then, thanks to Lemma 5 and (86), so we get

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This work was supported by the Postgraduate Innovation Team Project of Shaanxi Normal University (No. TD2020008Z) and the Recruitment Program for Young Professionals of China.

References

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Fuctions, CRC Press Taylor and Francis Group, Boca Raton, Second edition, 2015.
F. Maggi, “Sets of finite perimter and geometric variational problems,” An introduction to geometric measure theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, vol. 135, 2012.
View at: Google Scholar
T. Wang, “The affine Sobolev-Zhang inequality on ,” Advances in Mathematics, vol. 230, pp. 2457–2473, 2012.
View at: Google Scholar
H. Busemann, “The isoperimetric problem for Minkowski area,” American Journal of Mathematics, vol. 71, no. 4, p. 743, 1949.
View at: Publisher Site | Google Scholar
J. E. Brothers and F. Morgan, “The isoperimetric theorem for general integrands,” Michigan Mathematical Journal, vol. 41, no. 3, pp. 419–431, 1994.
View at: Publisher Site | Google Scholar
I. Fonseca, “The Wulff theorem revisited,” Proceedings: Mathematical and Physical Sciences, vol. 432, no. 1884, pp. 125–145, 1991.
View at: Google Scholar
I. Fonseca and S. Müller, “A uniqueness proof for the Wulff theorem,” Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol. 119, no. 1-2, pp. 125–136, 1991.
View at: Publisher Site | Google Scholar
R. J. Gardner, “The Brunn-Minkowski inequality,” Bulletin of the American mathematical society, vol. 39, no. 3, pp. 355–406, 2002.
View at: Publisher Site | Google Scholar
V. D. Milman and G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces. With an Appendix by M. Gromov. Number 1200 in Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1986.
P. M. Gruber, Convex and Discrete Geometry, Springer-Verlag, Berlin Heidelberg, 2007.
J. E. Taylor, “Crystalline variational problems,” Bulletin of the American Mathematical Society, vol. 84, no. 4, pp. 568–588, 1978.
View at: Publisher Site | Google Scholar
G. Wulff, “On the question of the rate of growth and the dissolution of the crystal surfaces, (German) Z,” Kristallografiya, vol. 34, pp. 440–530, 1901.
View at: Google Scholar
A. Figalli, F. Maggi, and A. Pratelli, “A mass transportation approach to quantitative isoperimetric inequalities,” Inventiones Mathematicae, vol. 182, no. 1, pp. 167–211, 2010.
View at: Publisher Site | Google Scholar
A. Figalli and Y. R. Zhang, “Strong stability for the Wulff inequality with a crystalline norm,” Communications on Pure and Applied Mathematics, vol. 75, no. 2, pp. 422–446, 2022.
View at: Publisher Site | Google Scholar
R. Neumayer, “A strong form of the quantitative Wulff inequality,” SIAM Journal on Mathematical Analysis, vol. 48, no. 3, pp. 1727–1772, 2016.
View at: Google Scholar
M. Perugini, Rigidity for Perimeter Inequalities under Symmetrization, University of Sussex, 2019, http://sro.sussex.ac.uk/id/eprint/84490/1/Perugini%2C%20Matteo.pdf.
E. De Giorgi, “On the isoperimetric property of the hypersphere, in the class of sets having an oriented boundary of finite measure, (Italian),” Michigan Mathematical Journal, vol. 5, 1958.
View at: Google Scholar
E. De Giorgi, “Selected papers,” Springer Collected Works in Mathematics, Springer, Heidelberg, 2013.
View at: Google Scholar
A. Cianchi, “Symmetrization in anisotropic elliptic problems,” Communications in Partial Differential Equations, vol. 32, no. 5, pp. 693–717, 2007.
View at: Publisher Site | Google Scholar
V. S. Klimov, “On the symmetrization of anisotropic integral functionals,” Russian Mathematics C/C of Izvestiia-Vysshie Uchebnye Zavedeniia Matematika, vol. 43, no. 8, 1999.
View at: Google Scholar
J. Van Schaftingen, “Anisotropic symmetrization,” Annales de l'IHP Analyse non linéaire, vol. 23, no. 4, pp. 539–565, 2006.
View at: Publisher Site | Google Scholar
R. J. Gardner, Geometric Tomography, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, New York, vol. 58, 2006.
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, New York, vol. 151, 2014.
G. Ewald, Combinatorial Convexity and Algebraic Geometry, Graduate Texts in Mathematics, Springer-Verlag, New York, vol. 168, 1996.
View at: Publisher Site
G. Zhang, “The affine Sobolev inequality,” Journal of Differential Geometry, vol. 53, no. 1, pp. 183–202, 1999.
View at: Publisher Site | Google Scholar
M. Amar and G. Bellettini, “A notion of total variation depending on a metric with discontinuous coefficients,” Annales de l'Institut Henri Poincaré C, Analyse non linéaire, vol. 11, no. 1, pp. 91–133, 1994.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Jin Dai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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