Abstract

An anisotropic convex Lorentz-Sobolev inequality is established, which extends Ludwig, Xiao, and Zhang’s result to any norm from Euclidean norm, and the geometric analogue of this inequality is given. In addition, it implies that the (anisotropic) Pólya-Szegö principle is shown.

1. Introduction

The classical Pólya-Szegö principle (see, e.g., [1, 2]) states that for the inequality holds for every , where denotes the set of functions on that are smooth and have compact support and is the standard Euclidean norm. Here denotes the Schwarz symmetrization of , that is, a function whose level sets have the same measure as the level sets of and are dilates of the Euclidean unit ball . It has important applications to a large class of variational problems in different areas, for example, isoperimetric inequalities, optimal forms of Sobolev inequalities, and sharp a priori estimates of solutions to second-order elliptic or parabolic boundary value problems.

An anisotropic version of the classical Pólya-Szegö principle has been proved in [3], where convex symmetrization of is involved, which states that if is an origin-symmetric compact convex set, then for the inequality holds for every , where is the Minkowski functional of  the polar body of  . Here denotes the convex symmetrization of , that is, a function whose level sets have the same measure as the level sets of and are dilates of the set . Obviously, (2) reduces to (1) when (see Section 2 for unexplained notation and terminology).

A new approach to understanding Pólya-Szegö principle was proposed recently by Lutwak et al. [4] and Zhang [5]. Instead of using the classical technique on level sets , their approach is using the convexification of level sets . This technique plays a fundamental role in the newly emerged affine Pólya-Szegö principle (see, e.g., [4–9]). Despite this progress, the study of the Pólya-Szegö principle by using this technique is vacancy. This is the motivation of the present paper. More precisely, we show the Pólya-Szegö principle from the Brunn-Minkowski theory, different from the known proofs of the Pólya-Szegö principle based on the geometric measure theory (see, e.g., [1–3, 10–16]).

In [17], Ludwig et al. proved the following convex Lorentz-Sobolev inequality (see Theorem  2 in [17]): if and , then where denotes the Lebesgue measure on with . This inequality has a geometric analogue, namely,  the following isoperimetric  inequality:  for , where is an origin-symmetric compact convex set in and is the surface area of .

In this paper we establish the following anisotropic convex Lorentz-Sobolev inequality.

Theorem 1. If , , and is an origin-symmetric convex body in , then with equality if and only if is a dilate of for almost every .

Similarly, our inequality (5) has a geometric analogue, namely,  the following   Minkowski  inequality, for , where , are origin-symmetric compact convex sets in and is the mixed volume of , .

When , from , (5) and (6) reduce to (3) and (4), respectively.

It is shown that our inequality (5) implies the anisotropic Pólya-Szegö principle (2) for in Theorem 5. Hence it is also true in Euclidean case; that is, (3) implies (1) for . The arguments after Theorem 5 yield the fact that the anisotropic Pólya-Szegö principle (2) is still true for if we use the solution to the even normalized Minkowski problem.

2. Background Material

2.1. Elements of the Brunn-Minkowski Theory

For later reference, we quickly recall in this subsection some background material from the Brunn-Minkowski theory of convex bodies. This theory has its origin in the work of Firey from the 1960s and has expanded rapidly over the last couple of decades (see, e.g., [4, 8, 18–33]).

A convex body is a compact convex set in which is throughout assumed to contain the origin in its interior. We denote by the space of convex bodies equipped with the Hausdorff metric. Each convex body is uniquely determined by its support function defined by Let denote the Minkowski functional of  ; that is, .

The polar set of is the convex body defined by If , then it follows from the definitions of support functions and Minkowski functionals, as well as the definition of polar body, that

For ,  , the Minkowski combination is the convex body defined by

The mixed volume of   is defined in [25] by In particular, for every convex body .

It was shown in [25] that, for all convex bodies , where and the measure on is the classical surface area measure of . Recall that, for a Borel set , is the -dimensional Hausdorff measure of the set of all boundary points of for which there exists a normal vector of belonging to .

Note that for all and convex bodies .

2.2. The Convex Symmetrization of Functions

Given any measurable function such that for every , its distribution function is defined by The decreasing rearrangement of is defined by The Schwarz symmetrization of is the function defined by where is the standard Euclidean norm.

For an origin-symmetric convex body , the convex symmetrization of with respect to is defined as follows: where is the Minkowski functional of , with being a dilate of so that . Note that ,  , and are equimeasurable; that is, Therefore, we have

We will frequently apply Federer’s co-area formula (see, e.g., [34, page 258]). We state a version which is sufficient for our purposes: if is Lipschitz and is measurable, then, for any Borel set , where denotes -dimensional Hausdorff measure.

2.3. The Convexification of Level Sets

Suppose . For each real  , define the level set By Sard’s theorem, for almost every , the boundary of is a smooth -dimensional submanifold of with everywhere nonzero normal vector .

Now, we explain the technique called the convexification of  level sets (see [17] for more details). Let , where is open, be locally Lipschitz; let ; and suppose   for almost everywhere on  . For , define the convexification of the level set as the unique origin-symmetric convex body such that for all even , where .

Thus, equality (24) holds for almost every if .

3. The Anisotropic Convex Lorentz-Sobolev Inequality

The following lemma can be proved in the spirit of [17, 31, 35](e.g., see Lemma  3 in [35]).

Lemma 2. If and are origin-symmetric convex bodies, then, for almost every and , is a dilate of and

Proof. Since is Lipschitz (and therefore differentiable almost everywhere) and on , then, for almost every , where is the outer unit normal vector of at the point . Note that , for almost every ; hence we have Since is Lipschitz, then, for almost every , the set is the boundary of a dilate of with nonvanishing normal . It follows from Sard’s theorem that Hence there exists a unique such that for almost every . Indeed, we have . Then by (24), (18), (9), and the fact that is homogeneous of degree and (27), we obtain that for almost every and even . Thus, the uniqueness of the solution of the even Minkowski problem [25] and (14) implies that where . Since for any and any , we have for almost every .

Recall that the Minkowski inequality [25] states the following.

Theorem 3. If and , then with equality if and only if  , are dilates when and if and only if , are homothetic when .

Now, we prove the anisotropic convex Lorentz-Sobolev inequality.

Proof of Theorem 1. Noting that , by the co-area formula (21), (24), (13), and (32), we have where on for almost every and the second equality holds since is an origin-symmetric and the support function of is homogeneous of degree 1.
Equality (5) follows from equality (32) and the fact that is origin-symmetric.

It is shown above, Proof of Theorem 1, that the Minkowski inequality (32) implies inequality (5).

In what follows we will show that the Minkowski inequality (32) can be easily deduced from the anisotropic convex Lorentz-Sobolev inequality (5) for by taking Indeed, as shown in [17, Lemma 8], and with . Hence where

4. The Pólya-Szegö Principle

The following theorem can be seen as a weak form of the Pólya-Szegö principle (2).

Theorem 4. If and , are origin-symmetric convex body such that is not a dilate of , then, for ,

Proof. Since is a dilate of for almost every by Lemma 2, then the Minkowski inequality (32) between and is strict for almost every . Combined with (25), it follows that

We are now in the position to prove the Pólya-Szegö principle (2).

Theorem 5. Suppose is an origin-symmetric convex bodies in . If , , then

Proof. It was shown in [4, (6.3)] that the following differential inequality holds: Integrating both sides of the inequality gives Noting that and Combined with (5), we obtain that By the homogeneous of in (43) and (40), we only need to consider . So it is sufficient to prove that
The last equality is shown in [3]. Now, we prove this equality by using Lemma 2. Together with the co-area formula (21), the equality (24), the definition of in Lemma 2, (13), and , we obtain where on for almost every . And the second equality holds since is origin-symmetric and the support function of is homogeneous of degree 1.