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Journal of Functional Analysis 265 (2013) 1728–1748

www.elsevier.com/locate/jfa

The affine Pólya–Szegö principle:Equality cases and stability

Tuo Wang

Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria

Received 10 December 2012; accepted 1 June 2013

Available online 12 June 2013

Communicated by K. Ball

Abstract

A Brothers–Ziemer type theorem for the affine Pólya–Szegö principle and a quantitative affine Pólya–Szegö principle are established.© 2013 The Author. Published by Elsevier Inc.

Keywords: Affine Pólya–Szegö; Stability; Functional Minkowski problem

1. Introduction

The classical Pólya–Szegö principle states that the Lp norm of the gradient of a function onRn does not increase under Schwarz radially decreasing symmetrization. It plays a fundamentalrole in the solution to a number of variational problems in different areas such as isoperimetricinequalities, optimal forms of Sobolev inequalities, and sharp a priori estimates of solutions tosecond-order elliptic or parabolic boundary value problems (see, for example, [36–38,54,55] andthe references therein).

A strengthened version of the classical Pólya–Szegö principle has been established byCianchi, Lutwak, Yang, and Zhang (see, e.g., [16,47,49]). The strengthened principle is dis-covered through the Lp Brunn–Minkowski theory and it is remarkable that it is invariant underthe action of the special linear group.

E-mail address: tuowang734@gmail.com.

Open access under CC BY license.

0022-1236 © 2013 The Author. Published by Elsevier Inc.http://dx.doi.org/10.1016/j.jfa.2013.06.001

Open access under CC BY license.

T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748 1729

To be precise, for f ∈ W 1,p(Rn), the usual Sobolev spaces, the Lp affine energy Ωp(f ) isdefined [16], by

Ωp(f ) = cn,p

( ∫Sn−1

‖Duf ‖−np du

)−1/n

,

where cn,p = (nωn)1/n(

nωnωp−12ωn+p−2

)1/p , ωn = πn/2

Γ (1+ n2 )

, ‖ ·‖p is the usual Lp norm on Rn and Du(f )

is the directional derivative of f defined by

Duf = u · ∇f.

Given f ∈ W 1,p(Rn), its distribution function μf : [0,∞) → [0,∞] is defined by

μf (t) = ∣∣{x ∈Rn:

∣∣f (x)∣∣ > t

}∣∣,where | ·| denotes Lebesgue measure on Rn. The decreasing rearrangement f ∗ : [0,∞) → [0,∞]of f is defined by

f ∗(s) = inf{t > 0: μf (t) � s

}.

The symmetric rearrangement of f is the function f � :Rn → [0,∞] defined by

f �(x) = f ∗(ωn|x|n),where | · | is the standard Euclidean norm.

For an origin-symmetric convex body K , the convex symmetrization f K of f with respect toK is defined as follows:

f K(x) = f ∗(ωn‖x‖n

K̃

),

where ‖ · ‖K̃ is the Minkowski functional of K̃ , with K̃ being a dilation of K so that |K̃| = ωn.The classical Pólya–Szegö principle could be stated as

‖∇f ‖p �∥∥∇f �

∥∥p, (1)

here ∇f is the weak derivative of f on Rn.And the affine Pólya–Szegö principle established by Cianchi, Lutwak, Yang and Zhang [51]

states that if f ∈ W 1,p(Rn), then

Ωp(f ) �Ωp

(f �

). (2)

The affine energy Ωp(·) is invariant under the action of the special linear group. And using theHölder inequality, one sees that the affine Pólya–Szegö principle is stronger than the Euclideancounterpart in the sense that

‖∇f ‖p �Ωp(f )� Ωp

(f �

) = ∥∥∇f �∥∥ . (3)

p

1730 T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748

As pointed out in [13], investigations on the uniqueness of the relevant extremals in Pólya–Szegö type principles are more recent, and typically require an additional delicate analysis. Sucha description has first been the object of the series of papers [7,26,36,57] and later has been re-cently extended and simplified by new contributions, including [8,9,12,14,17,18,20,22,21]. It isthe aim of this paper that we study the uniqueness of extremals in (2).

In the Euclidean case (1), partial results are contained in [36] and [57]. The problem was alsodiscussed by Friedman and Mcleod in [26] when f is of class Cn. However, as observed in [7],the proof in [26] contains an error which can be only repaired under additional assumptions.A general answer was given by Brothers and Ziemer [7], where the following theorem is proved.

Theorem 1. (See [7].) Let f be a compactly supported nonnegative function from W 1,p(Rn),1 < p < +∞, such that

∣∣{∣∣Df �∣∣ = 0

} ∩ (f �

)−1(0, ess supf )

∣∣ = 0. (4)

If ‖∇f �‖p = ‖∇f ‖p , then

f = f � a.e. in Rn,

up to a translation.

In [7], it is also shown that condition (4) is equivalent to the absolute continuity of μf . AndBrothers and Ziemer also constructed a smooth functions f , where ‖∇f ‖p = ‖∇f �‖p withoutf being any translate of f �, if one does not assume condition (4).

Thus in view of (3), the example constructed by Brothers and Ziemer would satisfy Ωp(f ) =Ωp(f �), but f fails to be any translate of symmetrization of f in any suitable sense. Thusabsolute continuity of μf is a necessary condition for our investigation of the extremalsof (2), and it turns out to be a sufficient condition! In particular we establish the followingtheorem.

Theorem 2. If f is a nonnegative function from W 1,p(Rn), 1 < p < +∞, such that

∣∣({∣∣∇f �∣∣ = 0

} ∩ {0 < f � < ess supf

})∣∣ = 0,

then

Ωp(f ) = Ωp

(f �

)if and only if there exists x0 ∈ Rn such that f (x) = f E(x + x0) a.e. in Rn, here E is an origin-centered ellipsoid in Rn.

The Euclidean Sobolev deficit of a function f ∈ W 1,p(Rn) is defined as the translation invari-ant quantity

De(f ) = ‖∇f ‖p

�− 1.

‖∇f ‖p

T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748 1731

Finding a quantitative version of (1) consists in proving that the “Euclidean Sobolev deficit”controls a suitable notion of “distance from the family of the extremal functions”. The followingEuclidean asymmetry of the function f is introduced in [13], where it is defined by

Ae(f ) = min± infx0∈Rn

∥∥f (·) ± f �(· + x0)∥∥

1,

and ‖ · ‖1 denotes the L1 difference. Note this asymmetry is also invariant under translations.It is established in [13] that

Ae(f ) � C[Mf

(De(f )r

) + De(f )]s

,

where C, r, s are positive constants depending on n and p, Mf is defined by

Mf (σ) = |({|∇f | � σ } ∩ {0 < |f | < ess sup |f |})||({|f | > 0})|

for σ � 0, f is normalized such that |{|f | > 0}| = 1 and ‖∇f ‖p = 1.We shall not attempt here to sketch the history and motivation of this problem, but simply

refer to [13] by Cianchi, Esposito, Fusco and Trombetti (and the references therein) where thisquantitative version of the Pólya–Szegö principle is established.

The main goal of the second part of this paper is to prove a quantitative version of the affinePólya–Szegö principle, with the assumption |{x ∈ Rn: |f (x)| > 0}| < ∞. To this purpose we in-troduce the following affine invariant quantity. For a Sobolev function f ∈ W 1,p(Rn), we definethe Lp affine ratio as

Ra(f ) := Ωp(f )

Ωp(f �).

Finding a quantitative version of (3) consists in proving that the “affine ratio” controls a moreusual notion of “affine distance from the family of the extremal functions”. To this aim is intro-duced the affine asymmetry of the function f , and it is defined by

Aa(f ) = min± infx0∈Rn,E∈E

∥∥f (·) ± f E(· + x0)∥∥

1,

where E is the family of all origin-centered ellipsoids. Note that the affine asymmetry is alsoinvariant under affine transformations.

We have the following result:

Theorem 3. If f is a function in W 1,p(Rn) with |{|f | > 0}| = 1 and ‖f ‖W 1,p(Rn) = 1, then

Aa(f ) � C

[Mf (σ) + σ + 1

σ

(Ra(f )p − 1

)r1

]r2

+ 2(er3

Ra(f )n−1Ra(f )n − 1

),

for every σ > 0, where C, r1, r2, r3 are positive constants depending only on n and p and‖f ‖W 1,p(Rn) is the usual Sobolev norm of f in W 1,p(Rn) defined by ‖f ‖W 1,p(Rn) = ‖f ‖p +‖∇f ‖p .

1732 T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748

2. Background material

2.1. Background on Lp Brunn–Minkowski theory

For quick reference we recall in this section some background material from the Lp Brunn–Minkowski theory of convex bodies. This theory has its origin in the work of Firey from the1960’s and has expanded rapidly over the last two decades since the work of Lutwak [44] (see,e.g., [10,11,31,32,35,39–42,44–46,51,47,49,48,50,58,59]).

A convex body is a compact convex subset of Rn with non-empty interior. We write Kn forthe set of convex bodies in Rn endowed with the Hausdorff metric and we denote by Kn

e the setof origin-symmetric convex bodies and by Kn

0 the set of convex bodies containing origin in theinterior. Each non-empty compact convex set K is uniquely determined by its support functionhK(·), defined by

hK(x) = max{x · y: y ∈ K}, x ∈ Rn,

where x · y denotes the usual inner product of x and y in Rn. Note that hK(·) is positivelyhomogeneous of degree one and sub-additive. Conversely, every function with these propertiesis the support function of a unique compact convex set.

The Banach–Mazur distance δBM(K,L) of the convex bodies K and L is defined by

δBM(K,L) = min{λ� 0: K − x ⊂ ψ(L − y) ⊂ eλ(K − x) for some ψ ∈ GL(n), x, y ∈R

n}.

In particular, if K and L are origin-symmetric, then x = y = 0 can be assumed.If K ∈ Kn

0 , then the polar body K◦ of K is defined by

K◦ = {x ∈ Rn: x · y � 1 for all y ∈ K

}.

From the polar formula for volume it follows that the n-dimensional Lebesgue measure |K◦| ofthe polar body K◦ can be computed by

∣∣K◦∣∣ = 1

n

∫Sn−1

hK(u)−n du,

where integration is with respect to spherical Lebesgue measure.For real p � 1 and α,β > 0, the Lp Minkowski combination of K,L∈Kn

0 is the convex bodyα · K +p β · L defined by

hα·K+pβ·L(·)p = αhK(·)p + βhL(·)p.

The Lp mixed volume Vp(K,L) of K,L ∈Kn0 was defined in [44] by

Vp(K,L) = plim+

|K +p ε · L| − |K|.

n ε→0 ε

T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748 1733

Clearly, the diagonal form of Vp reduces to ordinary volume, i.e., for K ∈ Kn0 ,

Vp(K,K) = |K|.It was also shown in [44] that for all convex bodies K,L ∈Kn

0 ,

Vp(K,L) = 1

n

∫Sn−1

hL(u)p dSp(K,u),

where dSp(K,u) = hK(u)1−p dS(K,u) and the measure S(K, ·) on Sn−1 is the classical surfacearea measure of K . Recall that for a Borel set ω ⊂ Sn−1, S(K,ω) is the (n − 1)-dimensionalHausdorff measure of the set of all boundary points of K for which there exists a normal vectorof K belonging to ω.

Projection bodies have become a central notion within the Brunn–Minkowski theory. Theyarise naturally in a number of different areas such as functional analysis, stochastic geometryand geometric tomography. The fundamental affine isoperimetric inequality which connects thevolume of a convex body with that of its polar projection body is the Petty projection inequality[53]. This inequality turned out to be far stronger than the classical isoperimetric inequality andhas led to Zhang’s affine Sobolev inequality [61].

The Lp projection body ΠpK of K ∈ Kn0 , was introduced in [45] as the convex body such

that

hΠpK(u)p =∫

Sn−1

|u · v|p dSp(K,v), u ∈ Sn−1.

The Lp analog of Petty’s projection inequality plays a key role in the Lp Brunn–Minkowskitheory. It was first proved by Lutwak, Yang, and Zhang [47] (see also Campi and Gronchi [10]for an independent approach): If K ∈ Kn

0 , then

|K|n/p−1∣∣Π◦

pK∣∣ � |B1|n/p−1

∣∣Π◦pB1

∣∣,where B1 is the unit ball and equality is attained if and only if K is an ellipsoid centered at theorigin.

The classical Minkowski problem also has a counterpart in the Lp Brunn–Minkowski theory.The even Lp Minkowski problem asks the following: Given an even Borel measure μ on Sn−1,does there exist a convex body K such that μ = Sp(K, ·)?

Lutwak [44] gave an affirmative answer to this problem when p �= n. The authors [50] intro-duced the volume-normalized Lp Minkowski problem, for which the case p = n can be handledas well. In particular, the following is proved:

Theorem 4. (See [50].) If 1 � p < ∞ and μ is an even Borel measure on the unit sphere Sn−1,then there exists a unique origin-symmetric convex body K such that

Sp(K, ·)|K| = μ

if and only if the support of μ is not contained in any (n − 1)-dimensional linear subspace.

1734 T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748

2.2. Anisotropic perimeter

We briefly recall that a Lebesgue set E of finite Lebesgue measure is said to be of finiteperimeter if its characteristic function χE belongs to BV (Rn), and then the Euclidean perimeterP(E) of E is given by the total variation of the distributional derivative of χE . Throughout thispaper, we shall refer to the monograph [2] for the basic properties of sets of finite perimeter.

The anisotropic isoperimetric inequality arises in connection with a natural generalization ofthe Euclidean notion of perimeter. If E is a set of finite perimeter in Rn, then its anisotropicperimeter, with respect to a convex body K ∈ Kn

0 , is defined as

PK(E) =∫

∂�E

hK

(νE(x)

)dHn−1(x),

where Hn−1 is the (n − 1)-dimensional Hausdorff measure, ∂�E denotes the reduced boundaryof E and νE : ∂�E → Sn−1 is the measure-theoretic outer unit normal vector field to E (see [2]for more details).

The anisotropic isoperimetric inequality states that,

PK(E) � n|K|1/n|E|1−1/n, (5)

for any E ⊂Rn of sets of finite perimeter.It was conjectured by Wulff [60] in 1901, that the equality holds if and only if E is homothetic

to K . And it was first shown that this is indeed the case by Taylor [56], and later, with an alterna-tive proof, by Fonseca and Müller [25]. Very recently, Figalli, Maggi and Pratelli [24] obtainedan optimal quantitative version of this inequality.

3. Extremal cases

As pointed out in [21], even if the method of Brothers and Ziemer is based on a geomet-rically clear approach, the rigorous justification of the arguments of [7] is accomplished afterovercoming serious technical difficulties by means of results from geometric measure theory.

The original proof of the affine Pólya–Szegö principle in [16] is based on the Lp projec-tion inequality applied to level sets of the Sobolev function. However, unlike the isoperimetricinequality, this inequality is invariant under all special linear transformations. This makes theBrothers–Ziemer type level set analysis much more complicated, since the method by Brothers–Ziemer depends ultimately on the homothety of the extremals in (5)!

We overcome this difficulty by using an approach by Lutwak, Yang and Zhang in [51], inparticular, the concept of the functional Minkowski problem on W 1,p(Rn). This will be thestarting point of the present work.

3.1. L1 case

As observed by Brothers and Ziemer [7], in fact, any Lipschitzian function f whose levelsets are (n − 1)-spheres satisfies

∫Rn |∇f �(x)|dx = ∫

Rn |∇f (x)|dx. For our investigation, weintroduce some facts that we will need.

T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748 1735

Theorem 5 (Functional Minkowski problem on W 1,1(Rn)). (See [51].) Given a function f ∈W 1,1(Rn), there exists a unique origin-symmetric convex body 〈f 〉 that

∫Rn

Φ(−∇f (x)

)dx =

∫Sn−1

Φ(u)dS(〈f 〉, u)

,

for every continuous function Φ : Rn → [0,∞) that is homogeneous of degree 1.

Theorem 6 (Co-area formula). If f ∈ W 1,1(Rn) and g :Rn → [0,+∞] is a Borel function, then

∫Rn

g(x)∣∣∇f (x)

∣∣dx =+∞∫

−∞

∫{f =t}

g(x)dHn−1(x) dt.

Theorem 7. (See [30,43,53].) If K is a convex body, then

|K|n−1∣∣Π◦

1 K∣∣ � |B1|n−1

∣∣Π◦1 B1

∣∣with equality if and only if K is an ellipsoid. Here Π◦

1 K stands for the polar projection body ofK and B1 is the unit ball of Rn.

Lemma 1. (See [7].) For a nonnegative f ∈ W 1,1(Rn), {x: f (x) = t} is the reduced boundaryof {x: f (x) > t} up to an Hn−1-negligible set for almost all t ∈ [0,∞).

Lemma 2. (See [19].) For positive f ∈ W 1,1(Rn) and almost all t ∈ [0,∞), −∇f (x)|∇f (x)| is the

measure-theoretic outer normal to {x: f (x) > t} for {x: f (x) = t} up to an Hn−1-negligibleset.

Theorem 8. For nonnegative f ∈ W 1,1(Rn), we have Ω1(f ) = Ω1(f�) if and only if the level

sets [f ]t = {x: f (x) � t} are homothetic ellipsoids up to an Ln-negligible set for almost allt ∈ [0,∞).

Proof. By Theorem 5, we have

∫Sn−1

Φ(u)dS(〈f 〉, u) =

∫Rn

Φ(∇f (x)

)dx

for any 1-homogeneous continuous function Φ : Rn → [0,∞).By the co-area formula, we have

∫n

Φ(∇f (x)

)dx =

+∞∫−∞

∫Φ

( ∇f (x)

|∇f (x)|)

dHn−1 dt.

R f =t

1736 T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748

By taking Φ = h〈f 〉, recalling Lemma 1, Lemma 2 and (5), we get

n∣∣〈f 〉∣∣ =

∫Sn−1

h〈f 〉(x) dS(〈f 〉, x)

=+∞∫0

P〈f 〉([f ]t

)dt

� n

+∞∫0

∣∣〈f 〉∣∣1/n∣∣[f ]t∣∣1/n′

dt.

Therefore, |〈f 〉|1/n′ �∫ +∞

0 |[f ]t |1/n′dt and there is equality if and only if [f ]t is homothetic

to 〈f 〉 for almost every t ∈R.Therefore, by Theorem 7, we have

Cn

∣∣Π◦〈f 〉∣∣−1/n �+∞∫0

∣∣[f ]t∣∣1/n′

dt, (6)

with equality if and only if [f ]t is homothetic to an ellipsoid for almost every t ∈ [0,∞), heren′ = n

n−1 , Cn is some optimal dimensional constant. We conclude the proof by noting that A1(f )

is the product of left hand side of (6) and a dimensional constant. �But notice that these ellipsoids need not to be concentric.

3.2. Lp case for p > 1

Theorem 9 (Functional Minkowski problem on W 1,p(Rn)). (See [51].) Given 1 < p < ∞ and afunction f ∈ W 1,p(Rn), there exists a unique origin-symmetric convex body 〈f 〉p that

∫Rn

Φp(−∇f (x)

)dx = 1

|〈f 〉p|∫

Sn−1

Φ(u)p dSp

(〈f 〉p,u),

for every continuous function Φ :Rn → [0,∞) that is homogeneous of degree 1.

Following the definition of Lutwak, Yang and Zhang [51], we define the normalized Lp pro-jection body Π̃pK of K by:

hp

Π̃pK(v) = 1

|K|∫

Sn−1

|u · v|p dSp(K,u).

We observe that Ωp(f ) = cn,pn− 1n (|Π̃◦

p〈f 〉p|)− 1n .

T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748 1737

The Lp Minkowski inequality reads as:

Theorem 10. (See Lutwak [44].) If 1 < p < ∞ and K,L are origin-symmetric convex bodiesin Rn, then

Vp(K,L) � |K|1−p/n|L|p/n.

And equality holds if and only if L = tK for some t > 0.

Convex symmetrization behaves well for the functional Minkowski problem:

Lemma 3. If f ∈ W 1,p(Rn) and K,L are origin-symmetric convex bodies, then 〈f K 〉p is adilate of K and ∣∣⟨f K

⟩p

∣∣ = ∣∣⟨f L⟩p

∣∣.Proof. Since hK◦(·) is a Lipschitzian function and hK◦(·) = 1 on ∂K , for almost all x ∈ ∂K ,

σK(x) = ∇hK◦(x)

|∇hK◦(x)| ,

where σK(x) is the outer unit normal vector of K at the point x. And from the definition of thepolar body, we have

hK

(σK(x)

) = 1

|∇hK◦(x)| .

Also, it is well known (see [7]) that f ∗ is locally absolutely continuous on (0,∞). If K isnormalized such that |K| = |B1|, we have, by the co-area formula applied to hK◦(·), that:∫

Rn

Φp(−∇f K(x)

)dx

=∫Rn

Φp((

f ∗)′(ωnhK◦(x)n

)nωnhK◦(x)n−1∇hK◦(x)

)dx

=∞∫

0

∫∂K

tn−1((−(f ∗)′(

ωntn))

nωntn−1)p

Φp

( ∇hK◦(x)

|∇hK◦(x)|)∣∣∇hK◦(x)

∣∣p−1dHn−1(x) dt

= (nωn)p

∞∫0

tnp+n−p−1((−f ∗)′(ωnt

n))p

dt

∫Sn−1

Φp(u)h1−p(K,u)dS(K,u)

= (nωn)p

∞∫0

tnp+n−p−1((−f ∗)′(ωnt

n))p

dt

∫Sn−1

Φp(u)dSp(K,u)

for any Φ that is homogeneous of degree 1.

1738 T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748

So we get

1

|〈f K 〉p|Sp

(⟨f K

⟩p,u

) = C(f )Sp(K,u),

where C(f ) = (nωn)p

∫ ∞0 tnp+n−p−1((−f ∗)′(ωnt

n))p dt .By comparing the homogeneity degrees of Sp and V , we get

∣∣⟨f K⟩p

∣∣ −pn−p = C(f )

nn−p |K|.

Since f cK = f K for any c ∈ (0,∞) and any K ∈Kn0 , we have

∣∣⟨f K⟩p

∣∣ −pn−p = ∣∣⟨f L

⟩p

∣∣ −pn−p . �

A norm is uniquely determined by its unit ball, which by definition must be an origin-symmetric convex body. Thus we identify a norm with the support function of an origin-symmetric convex body for convenience.

The Pólya–Szegö principle for general Minkowski functionals together with its extremal caseshave been established.

Theorem 11. (See [1,20,22].) Let K ∈ Kn0 . If f is a nonnegative function from W 1,p(Rn),

1 < p < ∞, such that

∣∣({∣∣∇f �∣∣ = 0

} ∩ {0 < f � < ess supf

})∣∣ = 0,

then

∫Rn

hpK

(∇f (x))dx �

∫Rn

hpK

(∇f K(x))dx,

and

∫Rn

hpK

(∇f (x))dx =

∫Rn

hpK

(∇f K(x))dx,

if and only if f = f K a.e. in Rn (up to translations).

Lemma 4. For f ∈ W 1,p(Rn) nonnegative, we have

〈f 〉p ⊆ ⟨f 〈f 〉p ⟩

p,

with 〈f 〉p = 〈f 〈f 〉p 〉p if and only if f = f K a.e. in Rn (up to translations) for some K ∈ Kne .

T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748 1739

Proof. By Theorem 9, we get, for some K ∈Kne , that

∫Rn

hpK(∇f )dx = n

|〈f 〉p|∣∣〈f 〉p,K

∣∣.By Theorem 11, we have

n

|〈f 〉p|Vp

(〈f 〉p,K)� n

|〈f K 〉p|Vp

(⟨f K

⟩p,K

)for K ∈ Kn

e , with equality if and only if f ≡ f K a.e. in Rn up to a translation.By choosing K = 〈f 〉p , we get

∣∣⟨f 〈f 〉p ⟩p

∣∣� Vp

(⟨f 〈f 〉p ⟩

p, 〈f 〉p

).

By Lemma 3, we have 〈f 〈f 〉p 〉p is homothetic to 〈f 〉p .Thus by Theorem 11, we conclude

〈f 〉p ⊆ ⟨f 〈f 〉p ⟩

p,

with 〈f 〉p = 〈f 〈f 〉p 〉p if and only if f = f 〈f 〉p almost everywhere (up to translations). �For our purpose, it is easier to deal with these normalized versions of Lp Petty projection

inequalities.

Theorem 12 (Lp projection inequality). (See [47].) If 1 < p < ∞ and K is a convex body, then

∣∣Π̃◦pK

∣∣ �mp,n|K|where

mp,n =[ √

πΓ (p+n

2 )

2Γ (n2 + 1)Γ (

p+12 )

]n/p

.

Equality holds if and only if K is an ellipsoid.

Now we give the proof of Theorem 2.

Proof of Theorem 2. By choosing Φ(x) = |v · x| in Theorem 9, we see that

Ωp(f ) = cn,p

∣∣Π̃◦p

(〈f 〉p)∣∣−1/n (7)

where cn,p is a suitably chosen constant depending only on n,p.By Lemma 3 and Lemma 4, we have 〈f 〈f 〉p 〉p is a dilate of 〈f 〉p and

⟨f 〈f 〉p ⟩ ⊇ 〈f 〉p.

p

1740 T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748

Therefore, by the homogeneity of the projection operator, we get

∣∣Π̃◦p

(〈f 〉p)∣∣−1/n �

∣∣Π̃◦p

(⟨f 〈f 〉p ⟩

p

)∣∣−1/n,

with equality if and only if f = f K , up to translations, for some K ∈ Kne .

By Lemma 3, we know

∣∣⟨f 〈f 〉p ⟩p

∣∣ = ∣∣⟨f �⟩p

∣∣.Since 〈f 〈f 〉p 〉p is a dilate of 〈f 〉p and 〈f �〉p is a dilate of B1. We conclude from Theorem 12that

∣∣Π̃◦p

(⟨f 〈f 〉p ⟩

p

)∣∣−1/n �∣∣Π̃◦

p

(⟨f �

⟩p

)∣∣−1/n,

with equality if and only if 〈f 〉p is an ellipsoid.Now we have

Ωp(f ) = Ωp

(f �

),

which forces f = f K up to some translations for some K ∈Kne and 〈f K〉p to be an ellipsoid.

Therefore, we have

f = f E almost everywhere

for some ellipsoid E in Rn, up to translations. �4. Stability

The study of the stability – an issue of special interest in modern analysis – of aforementionedvariation problems is of great interest and still not completely settled, although contributionshave been made as far as Sobolev, isoperimetric and isocapacitary type inequalities are concerned[3–6,15,13,19,23,24,27–29,33,34,52].

Böröczky has established the following quantitative version of the Lp projection inequality:

Theorem 13. (See [6].) For K ∈ Kn0 with δ = δBM(K,B1), we have

|Π̃◦pK|

|K| �(1 − γ · δ792n+840p

) |Π̃◦pB1|

|B1| ,

where γ > 0 depends on n.

Let K ∈Kn0 be a convex body such that a|x|� hK(x) � b|x|, where a, b ∈ (0,∞). We denote

the deficit DK(f ) of f by:

DK(f ) =∫Rn h

pK(∇f )∫

hp

(∇f K)− 1.

Rn K

T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748 1741

The Euclidean case of the following theorem is due to Cianchi, Fusco, Esposito and Trombetti[13], the general convex case is due to Esposito and Ronca [19].

Theorem 14. (See [13,19].) Let p > 1, n > 2 and K ∈ Kn0 . There exist positive constants

r1, r2, r3 and C depending only on p, n, a and b, such that for every f ∈ W 1,p(Rn) satisfy-ing |({|f > 0|})| < ∞,

min± infx0∈Rn

∫Rn

∣∣f (x) ± f K(x + x0)∣∣dx

� C∥∥hK(∇f )

∥∥p

∣∣({|f | > 0})∣∣1+ 1

n− 1

p

[Mf (σ) + DK(f )r1 + ‖hK(∇f )‖p

σ |({|f | > 0})| 1p

DK(f )r2

]r3

,

for every σ > 0.

The functional Minkowski problem has strong affine invariance properties.

Lemma 5. (See [51].) If f ∈ W 1,p(Rn) and τ be a translation in Rn, then

〈f ◦ τ 〉p = 〈f 〉p.

Lemma 6. (See [44].) If K,L ∈Kn0 and ψ ∈ SL(n), then

Vp(ψK,ψL) = Vp(K,L).

Lemma 7. (See [51].) If f ∈ W 1,p(Rn) and ψ ∈ SL(n), then

⟨f ◦ ψ−1⟩

p= ψ〈f 〉p.

Lemma 8. If f ∈ W 1,p(Rn) and ψ ∈ SL(n), then∫Rn

hψ〈f 〉p(∇(

f ◦ ψ−1))pdx =

∫Rn

h〈f 〉p (∇f )p dx.

Proof. By Theorem 9, Lemma 6, and Theorem 10, we have∫Rn

h〈f 〉p (∇f )p dx = 1

|〈f 〉p|∫

Sn−1

h〈f 〉p (u)p dSp

(〈f 〉p,u)

= n.

Similarly, we have by recalling Lemma 7 that∫Rn

hψ〈f 〉p(∇(

f ◦ ψ−1))pdx = n. �

1742 T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748

Lemma 9 (John’s lemma). If K is an origin-symmetric convex body in Rn, then there is anorigin-centered ellipsoid E such that the inclusions

E ⊆ K ⊆ nE

hold.

Lemma 10. Given a Sobolev function f ∈ W 1,p(Rn), let δs(K,B1) denote the normalized sym-metric difference between K and B1, given by δs(K,B1) = |K�B1||K| , where |K| = |B1|. Then wehave

(eδBM(〈f 〉p,B1)

)n − 1 � δs

(ψ 〈̃f 〉p ,B1

),

for some ψ ∈ SL(n).

Proof. Without loss of generality, assume that

δBM

(〈f 〉p,B1) = R,

where B1 ⊆ ψ〈f 〉p ⊆ eRB1 for some ψ ∈ SL(n).Therefore we have

δs

(ψ〈f 〉p, cB1

) = |ψ〈f 〉p � cB1||cB1| ,

where c > 1 is chosen such that |cB1| = |〈f 〉p|.Since ψ〈f 〉p � cB1 ⊆ eRB1 � B1, we have

δs

(ψ 〈̃f 〉p ,B1

) = δs

(ψ〈f 〉p, cB1

)�

(eR

)n − 1. �We are ready to establish the following quantitative estimate for the affine Pólya–Szegö prin-

ciple.

Proof of Theorem 3. By recalling Theorem 9, we find that the affine Pólya–Szegö principlecould be stated as:

cn,pn− 1n(∣∣Π̃◦

p〈f 〉p∣∣)− 1

n � cn,pn− 1n(∣∣Π̃◦

p

⟨f 〈f 〉p ⟩

p

∣∣)− 1n � cn,pn− 1

n(∣∣Π̃◦

p

⟨f �

⟩p

∣∣)− 1n .

Without loss of generality, we assume that the Banach–Mazur distance between 〈f 〉p and B

is attained by ln Rr

where

rψ〈f 〉p ⊂ B ⊂ Rψ〈f 〉p,

here 0 < r � R < ∞, ψ ∈ SL(n) and B is an origin-centered ball.

T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748 1743

And we assume

min± infx0∈Rn

∫Rn

∣∣f ◦ ψ−1(x) ± f ψ〈f 〉p (x + x0)∣∣dx =

∫Rn

∣∣f ◦ ψ−1(x) − f ψ〈f 〉p (x + τ0)∣∣dx

for convenience.Since Aa(·) is invariant under SL(n) and

⟨f ◦ ψ−1⟩

p= ψ〈f 〉p

for ψ ∈ SL(n), we could assume 〈f 〉p is in a position where there is an origin-centered ball B ′such that

B ′ ⊆ ψ〈f 〉p ⊆ nB ′.

From the triangle inequality, we have

Aa(f ) �∥∥f ◦ ψ−1(x) − f ψ〈f 〉p (x + τ0)

∥∥1 + ∥∥f ψ〈f 〉p (x + τ0) − f B(x + τ0)

∥∥1.

Therefore either

∥∥f ◦ ψ−1 − f ψ〈f 〉p∥∥1 �

1

2Aa(f )

or

∥∥f ψ〈f 〉p − f B∥∥

1 �1

2Aa(f ).

If ‖f ◦ ψ−1 − f ψ〈f 〉p‖1 � 12Aa(f ), by Lemma 4, we have

⟨f 〈f 〉p ⟩

p= (1 + c)〈f 〉p, (8)

where c � 0.We have

d|∇f | < h〈̃f 〉p (∇f ) < e|∇f |,

where d , e are dimensional constants, as B ⊆ 〈f 〉p ⊆ nB for some origin-centered ball B .Notice

Ra(f ) �|Π̃◦

p〈f 〉p|−1/n

|Π̃◦p〈f 〈f 〉p 〉p|−1/n

= 1 + c.

By the definition of 〈f 〉p , we have

∫n

h〈̃f 〉p (∇f )p dx = 1

|〈f 〉p|∫n−1

h〈̃f 〉p (u)p dSp

(〈f 〉p,u)

R S

1744 T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748

and

∫Rn

h〈̃f 〉p(∇f 〈f 〉p)p

dx = 1

|〈f 〈f 〉p 〉p|∫

Sn−1

h〈̃f 〉p (u)p dSp

(⟨f 〈f 〉p ⟩

p,u

).

Thus, using (8), we observe

∫Rn h〈̃f 〉p (∇f )p dx∫

Rn h〈̃f 〉p (∇f 〈f 〉p )p dx− 1 = (1 + c)p − 1 �

(Ra(f )

)p − 1.

Therefore, by Theorem 14, Lemma 8 and the Hölder Inequality, we have

infx0∈Rn

∫Rn

∣∣f ◦ ψ−1(x) − f ψ〈f 〉p (x + x0)∣∣dx

� C‖f ‖W 1,p(Rn)

∣∣({|f | > 0})∣∣1+ 1

n− 1

p

×[Mf (σ) + D〈f 〉p (f )r1 + ‖f ‖W 1,p(Rn)

σ |({|f | > 0})| 1p

D〈f 〉p (f )r2

]r3

� C‖f ‖W 1,p(Rn)

∣∣({|f | > 0})∥∥1+ 1

n− 1

p

×[Mf (σ) + ((

Ra(f ))p − 1

)r1 + ‖f ‖W 1,p(Rn)

σ |({|f | > 0})| 1p

((Ra(f )

)p − 1)r2

]r3

for every σ > 0, here C, r1, r2, r3 are constants depending only on n and p.On the other hand, using the layer-cake representation and the Fubini theorem, we have

∥∥f ψ〈f 〉p − f B∥∥

1 =∫Rn

∣∣∣∣∣∞∫

0

χ{f ψ〈f 〉p >t}(x) dt −∞∫

0

χ{f B>t}(x) dt

∣∣∣∣∣dx

�∫Rn

∞∫0

∣∣χ{f ψ〈f 〉p >t}(x) − χ{f B>t}(x)∣∣dt dx

=∞∫

0

|Et |dt,

where Et = {[f ψ〈f 〉p ]t � [f B ]t }.Since

∣∣[f ψ〈f 〉p ] ∣∣ = ∣∣[f B] ∣∣,

t t

T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748 1745

we have

∥∥f ψ〈f 〉p − f B∥∥

1 =∞∫

0

δs

([f ψ〈f 〉p ]

t,[f B

]t

)∣∣[f B]t

∣∣dt.

Since

δs

([f ψ〈f 〉p ]

t,[f B

]t

)is a constant for t > 0 a.e., we have

∥∥f ψ〈f 〉p − f B∥∥

1 � δs

([f ψ〈f 〉p ]

t,[f B

]t

)‖f ‖L1(Rn) � δs

( 〈̃f 〉p ,B1)‖f ‖L1(Rn).

Notice that, from Lemma 3, we have

∣∣⟨f 〈f 〉p ⟩p

∣∣ = ∣∣⟨f B⟩p

∣∣.And

Ωp(f ) = cn,pn− 1n(∣∣Π̃◦

p〈f 〉p∣∣)− 1

n ,

thus

(Ra(f )

)−n �|Π̃◦

p 〈̃f 〉p ||Π̃◦

pB1| � 1 − γn · δBM

( 〈̃f 〉p ,B1)792n+840p

.

Thus we have

δBM

( 〈̃f 〉p ,B1)�

(1 − (Ra(f ))−n

γn

) 1792n+840p

.

From Lemma 10, we have

δs

( 〈̃f 〉p ,B)� e

(1−(Ra(f ))−n

γn)

n792n+840p − 1.

Thus we have

∥∥f ψ〈f 〉p − f B∥∥

1 �(e(

1−(Ra(f ))−n

γn)

n792n+840p − 1

)‖f ‖1

�(e(

1−(Ra(f ))−n

γn)

n792n+840p − 1

)∣∣{|f | > 0}∣∣ p−1

p ‖f ‖W 1,p(Rn).

1746 T. Wang / Journal of Functional Analysis 265 (2013) 1728–1748

Thus, by triangle inequality, we have

Aa(f )� C‖f ‖W 1,p(Rn)

∣∣({|f | > 0})∣∣1+ 1

n− 1

p

×[Mf (σ) + ((

Ra(f ))p − 1

)r1 + ‖f ‖W 1,p(Rn)

σ |({|f | > 0})| 1p

((Ra(f )

)p − 1)r2

]r3

+ 2(e(

1−(Ra(f ))−n

γn)

n792n+840p − 1

)∣∣{|f | > 0}∣∣ p−1

p ‖f ‖W 1,p(Rn).

The theorem is proved once we recall that |{x: ∇f (x) �= 0}| = 1 and ‖f ‖W 1,p(Rn) = 1. �Corollary 1. Let f be a function from W 1,p(Rn),1 < p < +∞, such that |{|Df | = 0} ∩(f )−1(0, ess sup |f |)| = 0, and |{|f | > 0}| < ∞. If

Ωp

(f �

) = Ωp(f ),

then either

f = f � ◦ ψ

or

f = −f � ◦ ψ

for some ψ ∈ SL(n) a.e. up to a translation.

Proof. By taking Ra(f ) = 1 and choosing σ → 0 in the proof of Theorem 3, we are done. �Acknowledgments

The work of the author was supported by Austrian Science Fund (FWF) Project P25515-N25.

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