TSWJ The Scientific World Journal 1537-744X 2356-6140 Hindawi Publishing Corporation 10.1155/2014/875245 875245 Research Article The Pólya-Szegö Principle and the Anisotropic Convex Lorentz-Sobolev Inequality Liu Shuai He Binwu Seyedalizadeh Seyed S. 1 Department of Mathematics Shanghai University Shanghai 200444 China shu.edu.cn 2014 1572014 2014 11 05 2014 24 06 2014 24 06 2014 15 7 2014 2014 Copyright © 2014 Shuai Liu and Binwu He. 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.

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 p1 the inequality (1)Rn|f|pdxRn|f|pdx holds for every fC0(Rn), where C0(Rn) denotes the set of functions on Rn that are smooth and have compact support and |·| is the standard Euclidean norm. Here f denotes the Schwarz symmetrization of f, that is, a function whose level sets have the same measure as the level sets of f and are dilates of the Euclidean unit ball B. 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 , where convex symmetrization of f is involved, which states that if K is an origin-symmetric compact convex set, then for p1 the inequality (2)RnfKopdxRnfKKopdx holds for every fC0(Rn), where ·Ko is the Minkowski functional of  the polar body of  K. Here fK denotes the convex symmetrization of f, that is, a function whose level sets have the same measure as the level sets of f and are dilates of the set K. Obviously, (2) reduces to (1) when K=B (see Section 2 for unexplained notation and terminology).

A new approach to understanding Pólya-Szegö principle was proposed recently by Lutwak et al.  and Zhang . Instead of using the classical technique on level sets [f]t, their approach is using the Lp convexification of level sets ft. This technique plays a fundamental role in the newly emerged affine Pólya-Szegö principle (see, e.g., ). 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 Lp Brunn-Minkowski theory, different from the known proofs of the Pólya-Szegö principle based on the geometric measure theory (see, e.g., [13, 1016]).

In , Ludwig et al. proved the following convex Lorentz-Sobolev inequality (see Theorem  2 in ): if fC0(Rn) and 1p<n, then (3)Rn|f|pdxnκnp/n0V(ft)(n-p)/ndt, where V denotes the Lebesgue measure on Rn with κn=V(B)=πn/2/Γ(1+n/2). This inequality has a geometric analogue, namely,  the following Lp isoperimetric  inequality:  for 1<p<n, (4)Sp(L)nκnp/nV(L)(n-p)/n, where L is an origin-symmetric compact convex set in Rn and Sp(L) is the Lp surface area of L.

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

Theorem 1.

If fC0(Rn), 1pn, and K is an origin-symmetric convex body in Rn, then (5)RnfKopdxnV(K)p/n0V(ft)(n-p)/ndt with equality if and only if ft is a dilate of K for almost every t>0.

Similarly, our inequality (5) has a geometric analogue, namely,  the following  Lp Minkowski  inequality, for 1<p<n, (6)Vp(L,K)V(L)(n-p)/nV(K)p/n, where L, K are origin-symmetric compact convex sets in Rn and Vp(L,K) is the Lp mixed volume of L, K.

When L=B, from Sp(L)=nVp(L,B), (5) and (6) reduce to (3) and (4), respectively.

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

2. Background Material 2.1. Elements of the <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M65"><mml:mrow><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> Brunn-Minkowski Theory

For later reference, we quickly recall in this subsection some background material from the Lp 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, 1833]).

A convex body is a compact convex set in Rn which is throughout assumed to contain the origin in its interior. We denote by Kon the space of convex bodies equipped with the Hausdorff metric. Each convex body K is uniquely determined by its support function hK=h(K,·):RnR defined by (7)hK(x)=h(K,x):=max{x·y:yK}. Let ·K:Rn[0,) denote the Minkowski functional of  KKon; that is, xK=min{λ0:xλK}.

The polar set Ko of KKon is the convex body defined by (8)Ko={xRn:x·y1yK}. If KKon, then it follows from the definitions of support functions and Minkowski functionals, as well as the definition of polar body, that (9)hK(·)=h(K,·)=·Ko.

For p1,  K,LKon, the Lp Minkowski combination K+pL is the convex body defined by (10)h(K+pL,·)p=h(K,·)p+h(L,·)p.

The Lp mixed volume Vp(K,L) of  K,LKon is defined in  by (11)Vp(K,L)=pnlimε0+V(K+pε1/pL)-V(K)ε. In particular, (12)Vp(K,K)=V(K) for every convex body K.

It was shown in  that, for all convex bodies K,LKon, (13)Vp(K,L)=1nSn-1hLp(u)dSp(K,u), where Sp(K,u)=hK(u)1-pdS(K,u) and the measure S(K,·) on Sn-1 is the classical surface area measure of K. Recall that, for a Borel set ωSn-1, S(K,ω) is the (n-1)-dimensional Hausdorff measure of the set of all boundary points of K for which there exists a normal vector of K belonging to ω.

Note that (14)Sp(tK,·)=tn-pSp(K,·) for all t>0 and convex bodies K.

2.2. The Convex Symmetrization of Functions

Given any measurable function f:RnR such that V({xRn:|f(x)|>t})< for every t>0, its distribution function μf:[0,)[0,] is defined by (15)μf(t)=V({xRn:|f(x)|>t|f(x)|1}A1). The decreasing rearrangement f*:[0,)[0,] of f is defined by (16)f*(s)=inf{t0:μf(t)s}. The Schwarz symmetrization of f is the function f:Rn[0,] defined by (17)f(x)=f*(κn|x|n), where |·| is the standard Euclidean norm.

For an origin-symmetric convex body K, the convex symmetrization fK of f with respect to K is defined as follows: (18)fK(x)=f*(κnxK~n), where xK~ is the Minkowski functional of K~, with K~ being a dilate of K so that V(K~)=κn. Note that f,  f*, and fK are equimeasurable; that is, (19)μf=μf*=μfK. Therefore, we have (20)f=f*(0)=fK.

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 f:RnR is Lipschitz and g:Rn[0,) is measurable, then, for any Borel set AR, (21)f-1(A){|f|>0}g(x)dx=Af-1{t}g(x)|f(x)|dHn-1(x)dt, where Hn-1 denotes (n-1)-dimensional Hausdorff measure.

2.3. The <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M139"><mml:mrow><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> Convexification of Level Sets

Suppose fC0(Rn). For each real  t>0, define the level set (22)[f]t={xRn:|f(x)|t|f(x)|1}. By Sard’s theorem, for almost every t>0, the boundary (23)[f]t={xRn:|f(x)|=t|f(x)|1} of [f]t is a smooth (n-1)-dimensional submanifold of Rn with everywhere nonzero normal vector f(x).

Now, we explain the technique called the Lp convexification of  level sets (see  for more details). Let f:UR, where URn is open, be locally Lipschitz; let t>0; and suppose f(x)0  for almost everywhere on  [f]t={xU:|f(x)|=t}. For 1pn, define the Lp convexification ft of the level set [f]t as the unique origin-symmetric convex body such that (24)Sn-1φ(u)dSp(ft,uft1)=[f]tφ(ν(x))|f|p-1dHn-1(x) for all even φC(Sn-1), where ν(x)=-f(x)/|f(x)|.

Thus, equality (24) holds for almost every t>0 if fC0(Rn).

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 ).

Lemma 2.

If fC0(Rn) and K,L are origin-symmetric convex bodies, then, for almost every t(0,f) and 1pn, fKt is a dilate of K and (25)V(fKtfLt1)=V(fLtfLt1).

Proof.

Since hKo is Lipschitz (and therefore differentiable almost everywhere) and hKo(x)=1 on K, then, for almost every xK, (26)νK(x)=hKo(x)|hKo(x)|, where νK(x) is the outer unit normal vector of K at the point x. Note that hK(hKo(x))=1, for almost every xRn; hence we have (27)hK(νK(x))=1|hKo(x)|. Since fK is Lipschitz, then, for almost every t(0,f), the set [fK]t is the boundary of a dilate of K with nonvanishing normal fK. It follows from Sard’s theorem that (28)Hn({xRn:|f(x)|=t|f(x)|1}A1)=0foralmosteveryt>0. Hence there exists a unique s>0 such that t=f*(κnsn) for almost every t(0,f). Indeed, we have s=(μf(t)/κn)1/n. Then by (24), (18), (9), and the fact that hK~o is homogeneous of degree 0 and (27), we obtain that (29)Sn-1φp(u)dSp(fKt,ufKt1)=[fK]tφp(ν(x))|fK(x)|p-1dHn-1(x)=[fK]tφp(ν(x))lllll×|(f*)(κnhK~o(x)n)nκnhK~o(x)n-1hK~o(x)|p-1dHn-1(x)=sK~φp(ν(x))|(f*)(κnsn)nκnsn-1hK~o(x)|p-1dHn-1(x)=sn-1(-(f*)(κnsn)nκnsn-1)p-1×K~φp(νK~(x))|hK~o(x)|p-1dHn-1(x)=(μf(t)κn)(n-1)/n(-(f*)(μf(t))nκn(μf(t)κn)(n-1)/n)p-1×K~φp(νK~(x))hK~(νK~(x))1-pdHn-1(x)=np-1κn(p-n)/nμf(t)(n-1)p/n(-μf(t))1-p×Sn-1φp(u)dSp(K~,u) for almost every t(0,f) and even φC(Sn-1). Thus, the uniqueness of the solution of the even Lp Minkowski problem  and (14) implies that (30)fKt=c(f,t)1/(n-p)K~foralmosteveryt(0,f), where c(f,t)=np-1κn(p-n)/nμf(t)(n-1)p/n(-μf(t))1-p. Since fcK=fK for any c>0 and any KKon, we have (31)V(fKtfKt1)=V(fLtfLt1)=c(f,t)n/(n-p)κn for almost every t(0,f).

Recall that the Lp Minkowski inequality  states the following.

Theorem 3.

If p1 and L,KKon, then (32)Vp(L,K)V(L)1-p/nV(K)p/n with equality if and only if  L, K are dilates when p>1 and if and only if L, K are homothetic when p=1.

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

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

Noting that hK(·)=·Ko, by the co-area formula (21), (24), (13), and (32), we have (33)RnhK(f)pdx=0[f]thK(f)p1|f|dHn-1(x)dt=0[f]thK(ν(x))p|f|p-1dHn-1(x)dt=0Sn-1hK(u)pdSp(ft,uft1)dt=0nVp(ft,Kft1)dtnV(K)p/n0V(ftft1)(n-p)/ndt, where ν(x)=-f(x)/|f(x)| on [f]t for almost every t>0 and the second equality holds since K is an origin-symmetric and the support function of K is homogeneous of degree 1.

Equality (5) follows from equality (32) and the fact that ft is origin-symmetric.

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

In what follows we will show that the Lp Minkowski inequality (32) can be easily deduced from the anisotropic convex Lorentz-Sobolev inequality (5) for 1<p<n by taking (34)f(x)=g(xL),whereg(s)=(1+sp/(p-1))1-n/p. Indeed, as shown in [17, Lemma 8], (35)ft=cp(t)L, and cp(t)n-p=|g(s)|p-1sn-1 with t=g(s). Hence (36)iiiiiiiiiiRnfKopdx=0nVp(ft,Kft1)dtiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=0nVp(cp(t)L,K)dtlllllllllllllllllllllilllllllllll=nV(L,K)0cp(t)n-pdt,0V(ftft1)(n-p)/ndt=0V(cp(t)L)(n-p)/ndtiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=V(L)(n-p)/n0cp(t)n-pdt, where (37)0cp(t)n-pdt=(n-p)p(p-1)p-1pB(n-pp,np-n+pp).

4. The Pólya-Szegö Principle

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

Theorem 4.

If fC0(Rn) and K, L are origin-symmetric convex body such that L is not a dilate of K, then, for 1pn, (38)RnfLKopdx>RnfKKopdx.

Proof.

Since fLt is a dilate of L for almost every t(0,f) by Lemma 2, then the Lp Minkowski inequality (32) between fLt and K is strict for almost every t(0,f). Combined with (25), it follows that (39)RnfLKopdx=0nVp(fLt,KfLt1)dt>nV(K)p/n0V(fLtfLt1)(n-p)/ndt=nV(K)p/n0V(fKtfKt1)(n-p)/ndt=0nVp(fKt,KfKt1)dt=RnfKKopdx.

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

Theorem 5.

Suppose K is an origin-symmetric convex bodies in Rn. If fC0(Rn), 1pn, then (40)RnfKopdxRnfKKopdx.

Proof.

It was shown in [4, (6.3)] that the following differential inequality holds: (41)V(ftft1)(n-p)/nnp-1μf(t)(n-1)p/n(-μf(t))1-p. Integrating both sides of the inequality gives (42)0V(ftft1)(n-p)/ndtnp-10μf(t)(n-1)p/n(-μf(t))1-pdt. Noting that hK(·)=·Ko and Combined with (5), we obtain that (43)RnhK(f)pdxnpV(K)p/n0μf(t)(n-1)p/n(-μf(t))1-pdt. By the homogeneous of K in (43) and (40), we only need to consider V(K)=κn. So it is sufficient to prove that (44)RnhK(fK)pdx=npκnp/n0μf(t)(n-1)p/n(-μf(t))1-pdt.

The last equality is shown in . Now, we prove this equality by using Lemma 2. Together with the co-area formula (21), the equality (24), the definition of c(f,t) in Lemma 2, (13), and V(K)=κn, we obtain (45)RnhK(fK(x))pdx=0([fK]thK(fK(x))p|fK(x)|dHn-1(x))dt=0[fK]thK(ν(x))p|fK|p-1dHn-1(x)dt=0Sn-1hK(u)pdSp(fKt,ufKt1)dt=0Sn-1c(f,t)hK(u)pdSp(K,u)dt=0c(f,t)dtSn-1hK(u)pdSp(K,u)=nV(K)0np-1κn(p-n)/nμf(t)(n-1)p/n(-μf(t))1-pdt=npκnp/n0μf(t)(n-1)p/n(-μf(t))1-pdt, where ν(x)=-fK(x)/|fK(x)| on [fK]t for almost every t>0. And the second equality holds since K is origin-symmetric and the support function of K is homogeneous of degree 1.

Moreover, Theorem 5 can be proved for p1 by using the solution to the even normalized Lp Minkowski problem as in [7, 9]. More precisely, suppose fC0(Rn), for p1, and define the normalized Lp convexification ft~ as the unique origin-symmetric convex body such that (46)1V(ft~)  Sn-1g(u)dSp(ft~,u)=[f]tg(ν(x))|f|p-1dHn-1(x), for almost every t>0. By taking slight modifications in the proof of Theorem 1, we obtain (47)RnhK(f)pdxnV(K)p/n0V(ft~)-p/ndt. Similar to the proof of Theorem 5, together with the observation in [7, (4.22)] that (48)V(ft~)-p/nnp-1μf(t)(n-1)p/n(-μf(t))1-p, we also get (43). So Theorem 5 remains true for p=n.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All the authors contributed equally to the paper. All the authors read and approved the final paper.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (Grant no. 11371239), Shanghai Leading Academic Discipline Project (Project no. J50101), and the Research Fund for the Doctoral Programs of Higher Education of China (20123108110001).

Pólya G. Szegő G. Isoperimetric Inequalities in Mathematical Physics 1951 Princeton, NJ, USA Princeton University Press MR0043486 Talenti G. Best constant in Sobolev inequality Annali di Matematica Pura ed Applicata. Serie Quarta 1976 110 353 372 10.1007/BF02418013 MR0463908 ZBL0353.46018 2-s2.0-34250392866 Alvino A. Ferone V. Trombetti G. Convex symmetrization and applications Annales de l'Institut Henri Poincare C: Non Linear Analysis 1997 14 2 275 293 10.1016/S0294-1449(97)80147-3 MR1441395 2-s2.0-0001273694 Lutwak E. Yang D. Zhang G. Sharp affine Lp Sobolev inequalities Journal of Differential Geometry 2002 62 1 17 38 MR1987375 2-s2.0-0038381937 Zhang G. The affine Sobolev inequality Journal of Differential Geometry 1999 53 1 183 202 MR1776095 ZBL1040.53089 2-s2.0-0001846041 Alonso-Gutiérrez D. Bastero J. Bernués J. Factoring Sobolev inequalities through classes of functions Proceedings of the American Mathematical Society 2012 140 10 3557 3566 10.1090/S0002-9939-2012-11355-3 MR2929024 2-s2.0-84862865644 Cianchi A. Lutwak E. Yang D. Zhang G. Affine Moser-Trudinger and MORrey-Sobolev inequalities Calculus of Variations and Partial Differential Equations 2009 36 3 419 436 10.1007/s00526-009-0235-4 MR2551138 2-s2.0-70350389542 Haberl C. Schuster F. E. Asymmetric affine Lp Sobolev inequalities Journal of Functional Analysis 2009 257 3 641 658 10.1016/j.jfa.2009.04.009 MR2530600 2-s2.0-67349142409 Haberl C. Schuster F. Xiao J. An asymmetric affine Pólya-Szegö principle Mathematische Annalen 2012 352 3 517 542 10.1007/s00208-011-0640-9 MR2885586 Brothers J. E. Ziemer W. P. Minimal rearrangements of Sobolev functions Journal für die Reine und Angewandte Mathematik 1988 384 153 179 MR929981 Cianchi A. Esposito L. Fusco N. Trombetti T. A quantitative Pólya-Szegö principle Journal für die Reine und Angewandte Mathematik 2008 614 153 189 10.1515/CRELLE.2008.005 MR2376285 2-s2.0-39649092311 Cianchi A. Fusco N. Functions of bounded variation and rearrangements Archive for Rational Mechanics and Analysis 2002 165 1 1 40 10.1007/s00205-002-0214-9 MR1947097 ZBL1028.49035 2-s2.0-0036027235 Esposito L. Ronca P. Quantitative Pólya-Szegö principle for convex symmetrization Manuscripta Mathematica 2009 130 4 433 459 10.1007/s00229-009-0297-9 MR2563145 2-s2.0-70350743013 Esposito L. Trombetti C. Convex symmetrization and Pólya-Szegö inequality Nonlinear Analysis: Theory, Methods & Applications 2004 56 1 43 62 10.1016/j.na.2003.07.010 MR2031435 2-s2.0-0345356496 Ferone A. Volpicelli R. Minimal rearrangements of Sobolev functions: a new proof Annales de l'Institut Henri Poincare (C): Non Linear Analysis 2003 20 2 333 339 10.1016/S0294-1449(02)00012-4 MR1961519 2-s2.0-0345529248 Ferone A. Volpicelli R. Convex rearrangement: equality cases in the Pólya-Szegö inequality Calculus of Variations and Partial Differential Equations 2004 21 3 259 272 10.1007/s00526-003-0256-3 MR2094322 Ludwig M. Xiao J. Zhang G. Sharp convex Lorentz-Sobolev inequalities Mathematische Annalen 2011 350 1 169 197 10.1007/s00208-010-0555-x MR2785767 ZBL1220.26020 2-s2.0-79953242509 Campi S. Gronchi P. The Lp-Busemann-Petty centroid inequality Advances in Mathematics 2002 167 1 128 141 10.1006/aima.2001.2036 MR1901248 2-s2.0-0037091874 Campi S. Gronchi P. On the reverse Lp-Busemann-Petty centroid inequality Mathematika 2002 49 1-2 1 11 (2004) 10.1112/S0025579300016004 MR2059037 Haberl C. Schuster F. E. General Lp affine isoperimetric inequalities Journal of Differential Geometry 2009 83 1 1 26 MR2545028 2-s2.0-70349798879 Hug D. Lutwak E. Yang D. Zhang G. On the Lp Minkowski problem for polytopes Discrete & Computational Geometry 2005 33 4 699 715 10.1007/s00454-004-1149-8 MR2132298 2-s2.0-17444416723 Ludwig M. Ellipsoids and matrix-valued valuations Duke Mathematical Journal 2003 119 1 159 188 10.1215/S0012-7094-03-11915-8 MR1991649 ZBL1033.52012 2-s2.0-0042575606 Ludwig M. Minkowski valuations Transactions of the American Mathematical Society 2005 357 10 4191 4213 10.1090/S0002-9947-04-03666-9 MR2159706 2-s2.0-26444584569 Ludwig M. Reitzner M. A classification of SL(n) invariant valuations Annals of Mathematics 2010 172 2 1219 1267 10.4007/annals.2010.172.1223 MR2680490 2-s2.0-77957697204 Lutwak E. The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem Journal of Differential Geometry 1993 38 1 131 150 MR1231704 Lutwak E. The Brunn-Minkowski-Firey theory. {II}. Affine and geominimal surface areas Advances in Mathematics 1996 118 2 244 294 10.1006/aima.1996.0022 MR1378681 ZBL0853.52005 2-s2.0-0030600832 Lutwak E. Yang D. Zhang G. L p affine isoperimetric inequalities Journal of Differential Geometry 2000 56 1 111 132 MR1863023 2-s2.0-0003157976 Lutwak E. Yang D. Zhang G. A new ellipsoid associated with convex bodies Duke Mathematical Journal 2000 104 3 375 390 10.1215/S0012-7094-00-10432-2 MR1781476 2-s2.0-0034664274 Lutwak E. Yang D. Zhang G. On the Lp-Minkowski problem Transactions of the American Mathematical Society 2004 356 11 4359 4370 10.1090/S0002-9947-03-03403-2 MR2067123 2-s2.0-7544248811 Lutwak E. Yang D. Zhang G. L p John ellipsoids Proceedings of the London Mathematical Society 2005 90 2 497 520 10.1112/S0024611504014996 MR2142136 2-s2.0-28244435144 Lutwak E. Yang D. Zhang G. Optimal Sobolev norms and the Lp Minkowski problem International Mathematics Research Notices 2006 2006 21 10.1155/IMRN/2006/62987 MR2211138 2-s2.0-33645163698 Werner E. Ye D. New Lp affine isoperimetric inequalities Advances in Mathematics 2008 218 3 762 780 10.1016/j.aim.2008.02.002 MR2414321 2-s2.0-41849095075 Werner E. Ye D. Inequalities for mixed p-affine surface area Mathematische Annalen 2010 347 3 703 737 10.1007/s00208-009-0453-2 MR2640049 ZBL1192.52013 2-s2.0-77952097913 Federer H. Geometric Measure Theory 1969 Berlin, Germany Springer MR0257325 Wang T. The affine Pólya-Szegö principle: equality cases and stability Journal of Functional Analysis 2013 265 8 1728 1748 10.1016/j.jfa.2013.06.001 MR3079233