aa r X i v : . [ m a t h . M G ] S e p ON THE DISCRETE FUNCTIONAL L p MINKOWSKIPROBLEM
TUO WANG
Abstract.
The discrete functional L p Minkowski problem is posedand solved. As a consequence, the general affine P´olya-Szeg¨o prin-ciple and the general affine Sobolev inequalities are established.
Introduction
The even functional Minkowski problem was first introduced by Lut-wak, Yang and Zhang in [20] to answer the following question:Given a function f ∈ W , ( R n ), where W , ( R n ) is the usual L Sobolev space on R n , which norm k · k o on R n minimizes the quantity R R n k∇ f k o dx, if the unit ball of the dual norm of k · k o has the samevolume as the Euclidean unit ball in R n ?The solution to this problem turns out to be unique and is called theoptimal Sobolev norm.To answer the question, Lutwak, Yang and Zhang introduced thefollowing even functional Minkowski problem on W , ( R n ) :Given f ∈ W , ( R n ), find an o -symmetric convex body h f i e suchthat(1) Z S n − Ψ( u ) dS ( h f i e , u ) = Z R n Ψ( ∇ f ( x )) dx for all continuous functions Ψ on R n that are even and positivelyhomogeneous of degree 1. Here S ( h f i e , · ) is the Alexandrov-Fenchel-Jessen surface area measure of h f i e and o -symmetric stands for origin-symmetric.The optimal Sobolev body h·i e defined through equation (1) providesthe solution of the optimal Sobolev norm problem. The even functionalMinkowski problem is shown to play an important role in the theory ofaffine Sobolev inequalities on R n (see [13][20][27][29]). Using valuationson Sobolev spaces, Ludwig obtained a natural description of the oper-ator f
7→ h f i e (see [13]). By extending the even functional Minkowski problem to BV ( R n ) , the space of functions of bounded variations, theauthor was able to extend the Zhang-Sobolev inequality to BV ( R n ) andgot a characterization of the equality cases (see [27]). By combiningthe even functional Minkowski problem with the convex symmetriza-tion technique introduced in [1], the author got a Brothers-Ziemer typetheorem for the affine P´olya-Szeg¨o principle, which clarified the equal-ity condition of the affine P´olya-Szeg¨o principle (see [29]). Based onthese facts, it is natural to ask the following question:Is there always a hidden convex body that plays an important rolein each affine Sobolev type inequality?While for the symmetric optimal L p affine Sobolev inequality, theanswer was obtained by the Lutwak, Yang and Zhang [20]. The aim ofthis paper is to define and solve the general functional L p Minkowskiproblems on Sobolev spaces. By making use of the solutions of thegeneral functional L p Minkowski problems and the general L p affineisoperimetric inequalities [7], we arrive at the general affine P´olya-Szeg¨oprinciple and general affine Sobolev type inequalities, which include theresults in [8][9][22] as special cases. This shows that the crucial convexbody behind the discrete version general affine Sobolev is the solutionof the discrete functional Minkowski problem.To to specific, we study the discrete versions of functional L p MinkowskiProblems. Instead of W , ( R n ), we consider the functional L p Minkowskiproblem on L , ( R n ), a dense subspace of W ,p ( R n ). Definition 1.
The space of piecewise affine functions on R n , where afunction f : R n → R is called piecewise affine, if f is continuous andthere are finitely many n -dimensional simplices T , ..., T m ⊂ R n withpairwise disjoint interiors such that the restriction of f to each T i isaffine and outside T ∪ · · · ∪ T m . In particular, we get the following result:
Theorem 1.
Given a nontrivial function f ∈ L , ( R n ) , there exists aunique polytope h f i p ∈ P n such that (2) n Z R n Ψ p ( −∇ f ( x )) dx = Z S n − Ψ( u ) p dS p ( h f i p , u ) , for every continuous function Ψ : R n → [0 , ∞ ) that is homogeneous ofdegree . The new functional L p Minkowski problem enjoys some natural affineinvariance property and valuation property. By studying this new func-tional Minkowski problem on L , ( R n ), we are led to the general affineP´olya-Szeg¨o principle and general affine Sobolev inequalities. N THE DISCRETE FUNCTIONAL L p MINKOWSKI PROBLEM 3
We remark that, although the discovery of the discrete L p Minkowskiproblem on W ,p ( R n ) is the main issue of this piece of work, a veryimportant tool that we make use in this piece of work is the Haberl-Schuster version of the general L p affine isoperimetric inequality in [7].2. Background material
Background on the L p Brunn-Minkowski theory. For quick referencewe recall in this section some background material from the L p Brunn-Minkowski theory of convex bodies. This theory has its origin in thework of Firey from the 1960’s and has expanded rapidly over the lasttwo decades since the work of Lutwak [17] (see, e.g.,[2, 3, 4, 7, 9, 10,11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28]).A convex body is a compact convex subset of R n with non-emptyinterior. We write K n for the set of convex bodies in R n endowed withthe Hausdorff metric and we denote by K ne the set of origin symmetricconvex bodies, by K n the set of convex bodies containing origin in theinterior and by P n the set of polytopes containing origin in the interior.Each non-empty compact convex set K is uniquely determined by itssupport function h ( K, · ), defined by h ( K, x ) = max { x · y : y ∈ K } , x ∈ R n , where x · y denotes the Euclidean inner product of x and y in R n . Notethat h ( K, · ) is positively homogeneous of degree 1 and sub-additive.Conversely, every function with these two properties is the supportfunction of a unique compact convex set.For later use, we include the following lemma from [26]: Lemma 1.
Let K i and K be convex bodies in R n . If h ( K i , · ) → h ( K, · ) pointwise, then h ( K i , · ) → h ( K, · ) uniformly on S n − . If K ∈ K n , then the polar body K ◦ of K is defined by K ◦ = { x ∈ R n : x · y ≤ f or all y ∈ K } . From the polar formula for volume it follows that the n -dimensionalLebesgue measure | K ◦ | of the polar body K ◦ can be computed by | K ◦ | = 1 n Z S n − h ( K, u ) − n du, where integration is with respect to spherical Lebesgue measure.For real p ≥ α, β >
0, the L p Minkowski combination of
K, L ∈ K n is the convex body α · K + p β · L defined by h p ( α · K + p β · L, u ) = h p ( K, u ) + h p ( L, u ) , and the L p mixed volume of K, L is defined by
TUO WANG V p ( K, L ) = pn lim ǫ → + | K + p ǫ · L | − | K | ǫ . Clearly, the diagonal form of V p reduces to ordinary volume, i.e., for K ∈ K n , V p ( K, K ) = | K | . It was also shown in [17] that for all convex bodies
K, L ∈ K n ,V p ( K, L ) = 1 n Z S n − h ( L, u ) p dS p ( K, u ) , where dS p ( K, u ) = h ( K, u ) − p dS ( K, u ). Recall that for a Borel set ω ⊂ S n − , S ( K, ω ) is the ( n − H n − of the set of all boundary points of K for which there exists a normalvector of K belonging to ω. A fundamental inequality that we need is the L p Minkowski inequal-ity [17].
Theorem 2 ([17]) . If < p < ∞ and K, L ∈ K n , then (3) V p ( K, L ) ≥ | K | − p/n | L | p/n . Equality holds if and only if L = tK for some t > . Projection bodies have become a central notion within the Brunn-Minkowski theory. They arise naturally in a number of different areassuch as functional analysis, stochastic geometry and geometric tomog-raphy. The fundamental affine isoperimetric inequality which connectsthe volume of a convex body with that of its polar projection body isthe Petty projection inequality [25]. This inequality turned out to bestronger than the classical isoperimetric inequality and its functionalform is Zhang’s discovery, namely the affine Zhang-Sobolev inequality[30].The L p projection body Π p K of K ∈ K n , was introduced in [18] asthe convex body such that h (Π p K, u ) p = Z S n − | u · v | p dS p ( K, v ) , u ∈ S n − . The L p analog of Petty’s projection inequality plays a key role inthe L p Brunn-Minkowski theory. It was first proved by Lutwak, Yang,and Zhang [21] (see also Campi and Gronchi [3] for an independentapproach): If K ∈ K n , then | K | n/p − | Π p K | ≤ | B | n/p − | Π p B | , where B is the Euclidean unit ball and equality is attained if and onlyif K is an ellipsoid centered at the origin. N THE DISCRETE FUNCTIONAL L p MINKOWSKI PROBLEM 5
For a finite Borel measure µ on S n − , we define a continuous function C + p ( µ ) on S n − , the asymmetric L p cosine transform of µ , by( C + P µ )( u ) = Z S n − ( u · v ) p + dµ ( v ) , u ∈ S n − , where ( u · v ) + = max { u · v, } . The asymmetric L p projection bodyΠ p + ( K ) of K ∈ K n , first considered in [12], is the convex body definedby h (Π p + ( K ) , · ) p = C + p ( S p ( K, · )) . For p >
1, by using valuations, Ludwig [12] established the L p ana-logue of her classification of the projection operator: She showed thatthe convex bodies(1 − λ ) · Π + p ( K ) + p λ · Π − p ( K ) , K ∈ K n where 0 ≤ λ ≤ − p ( K ) = Π + p ( − K ), constitute all natural L p extensions of projection bodies.Haberl and Schuster established the following general L p Petty pro-jection inequalities in [7].
Theorem 3 ([7]) . Let K ∈ K n and p > . If Φ λ,p K is the convex bodydefined by (4) Φ λ,p ( K ) = (1 − λ ) · Π + p ( K ) + p λ · Π − p ( K ) , where ≤ λ ≤ , then | K | n/p − | Φ λ,p K | ≤ | B | n/p − | Φ λ,p B | , with equality if and only if K is an ellipsoid centered at the origin. The classical Minkowski problem has a counterpart in the L p Brunn-Minkowski theory. The L p Minkowski problem asks the following:Given a Borel measure µ on S n − , does there exist a convex body K such that µ = S p ( K, · )?If the data µ is an even measure, Lutwak [17] gave an affirmativeanswer to this problem when p = n . Later on, Lutwak, Yang and Zhang[24] introduced the volume-normalized even L p Minkowski problem, forwhich the case p = n can be handled as well. What we need in thiswork is the following discrete version of the L p Minkowski problem.
Theorem 4 ([10]) . Let vectors u , ..., u m ∈ S n − that are not containedin a closed hemisphere and real numbers α , ..., α m > be given. Then,for any p > with p = n , there exists a unique polytope P ∈ P n suchthat m X j =1 α j δ u j = S p ( P, · ) . TUO WANG
For treating the case p = n , we need the normalized version of theabove theorem. Theorem 5 ([10]) . Let vectors u , ..., u m ∈ S n − that are not containedin a closed hemisphere and real numbers α , ..., α m > be given. Thenthere exists a unique polytope P ∈ P n such that m X j =1 α j δ u j = S n ( P, · ) | P | . The discrete functional L p Minkowski problem
We assume 1 < p < ∞ and p = n throughout this section.3.1. The discrete case.Theorem 6.
Given a nontrivial function f ∈ L , ( R n ) , there exists aunique polytope h f i p ∈ P n such that (5) n Z R n Ψ p ( −∇ f ( x )) dx = Z S n − Ψ( u ) p dS p ( h f i p , u ) , for every continuous function Ψ : R n → [0 , ∞ ) that is homogeneous ofdegree .Proof. Let Σ = { x : ∇ f ( x ) = 0 } . Then the mapΥ n Z R n \ Σ Υ( − ∇ f ( x ) |∇ f ( x ) | ) |∇ f ( x ) | p dx defines a nonnegative bounded linear functional on the space of con-tinuous functions on S n − . It follows from the Riesz representationtheorem that there exists a unique Borel measure S p ( f, · ) on S n − suchthat n Z R n \ Σ Υ( − ∇ f ( x ) |∇ f ( x ) | ) |∇ f ( x ) | p dx = Z S n − Υ( u ) dS p ( f, u ) , for each continuous function Υ : S n − → R . N THE DISCRETE FUNCTIONAL L p MINKOWSKI PROBLEM 7
Choose Υ = Ψ p and radially extend Ψ to R n by requiring Ψ to be ofhomogeneity degree 1. Then we have n Z R n Ψ p ( −∇ f ( x )) dx = n Z R n \ Σ Ψ p ( −∇ f ( x )) dx = n Z R n \ Σ Ψ p ( − ∇ f ( x ) |∇ f ( x ) | ) |∇ f ( x ) | p dx = Z S n − Ψ p ( u ) dS p ( f, u ) . We claim that S p ( f, u ) is a finite combination of Dirac measures onthe sphere for any f ∈ L , ( R n ).Notice from the definition, we have S p ( f ( x ) , · ) = S p ( f ( x + x ) , · ) , for any x ∈ R n . Next we consider a triangulation of the support of f , i.e. on eachsimplex M i , f is affine, where i = 1 , ..., m. Without loss of generality, we assume f ( x ) = x · u i + c i for x ∈ M i . Then we have n Z R n Ψ p ( −∇ f ( x )) dx = n m X i =1 Z M i Ψ p ( −∇ f ( x )) dx = n m X i =1 | M i | Ψ p ( − u i ) dx. Therefore we have S p ( h f i p , · ) = n m X i =1 | M i | δ − u i , where δ − u i is the Dirac measure concentrated on − u i . We remark that the fact that S p ( f, u ) does not concentrate on anygreat sub-sphere on S n − follows basically from the same argument ofLemma 4.1 in [20]. In fact, by tracing back to the definition, one gets S p ( f, u )( S n − ) = 0 , where S n − is any big sphere of S n − . TUO WANG
Therefore, for any f ∈ L , ( R n ), S p ( f, u ) is a linear combination ofthe Dirac measures on the sphere which do not concentrate on anygreat subsphere. By using Theorem 4, we prove the existence of thepolytope h f i p ∈ P n . The uniqueness is a consequence of the L p Minkowski inequality(Theorem 2). (cid:3)
For P ∈ P n , define the piecewise affine function l P by requiringthat l P (0) = 1 , that l P ( x ) = 0 for x P , and that l P is affine oneach simplex with apex at the origin and base equal to a facet of P .Define P ,p ⊂ L , ( R n ) as the set of all l P for P ∈ P n . Note that for φ ∈ GL ( n ) , l φP = l P ◦ φ − . We remark that multiples and translates of l P ∈ P ,p ( R n ) correspondto linear elements within the theory of finite elements. Lemma 2.
For P ∈ P n , h l P i p = P. Proof.
Let P have facets F , ..., F m . For the facet F i , let u i be its unitouter normal vector and T i the convex hull of F i and the origin. Sincefor x ∈ T i , l P ( x ) = − u i h ( P, u i ) · x + 1and ∇ l P ( x ) = − u i h ( P, u i ) , we have that n Z R n Ψ p ( −∇ l P ( x )) dx = n m X i =1 Z T i Ψ p ( −∇ l P ( x )) dx = n m X i =1 Z T i Ψ p ( u i h ( P, u i ) ) dx = n m X i =1 Ψ p ( u i ) V n ( T i ) h ( P, u i ) p = m X i =1 Ψ p ( u i ) V n − ( F i ) h ( P, u i ) − p = Z S n − Ψ p ( u ) dS p ( P, u ) . Since S p ( P, · ) = S p ( h P i p , · ) , we have P = h P i p from Theorem 4. (cid:3) N THE DISCRETE FUNCTIONAL L p MINKOWSKI PROBLEM 9
Lemma 3. If f ∈ L , ( R n ) and φ ∈ SL ( n ) , then h f ◦ φ − i p = φ h f i p . Proof.
It is well-known (e.g. see [20]) that if V p ( φK, L ) = V p ( J, φ − L ) f or all convex bodies L ∈ K n , then K = J. Since by the affine invariance of the L p mixed volume, we have V p ( φ h f i p , L ) = V p ( h f i p , φ − L ) , and we know, by the change of variable y = φ ( x ), that Z R n h ( L, ∇ ( f ◦ φ − )( u )) p dy = Z R n h ( L, φ − t ∇ f ( φ − y )) p dy = Z R n h ( φ − L, ∇ f ( x )) p dx. Therefore, we have that V p ( h f ◦ φ − i p , L ) = V p ( h f i p , φ − L ) , which im-plies that h f ◦ φ − i p = φ h f i p . (cid:3) As a quick consequence of Lemma 3, we have the following corollary.
Corollary 1.
Given φ ∈ SL ( n ) and f ∈ L , ( R n ) , we have |h f i p | = |h f ◦ φ − i p | . The operator h·i p : L , ( R n ) → P n has the valuation property. Definition 2.
For two polytopes
K, L ∈ P n , we define the L p -Blaschkesum K♯ p L of K, L by requiring (6) S p ( K♯ p L, · ) = S p ( K, · ) + S p ( L, · ) . Remark: By Theorem 4, we know there is a polytope K♯ p L uniquelydefined through equation (6).For f, g ∈ L , ( R n ), the function f ∨ g denotes the pointwise maxi-mum and the function f ∧ g the pointwise minimum of f and g . Theorem 7.
The operator h·i p : ( L , ( R n ) , ∧ , ∨ ) → ( P n , ♯ p ) is an L p -Blaschke valuation.Proof. What we need to show is that Z S n − Ψ( u ) p dS p ( h f i p , u ) + Z S n − Ψ( u ) p dS p ( h g i p , u )= Z S n − Ψ( u ) p dS p ( h f ∧ g i p , u ) + Z S n − Ψ( u ) p dS p ( h f ∨ g i p , u ) , for every continuous Ψ : S n − → R and f, g ∈ L , ( R n ) . For this we triangulate the support of f and g so that f ( x ) = x · u i + c i on P i for i = 1 , ..., m and g ( x ) = x · v i + d i on Q i for i = 1 , ..., n , andwithout loss of generality, we assume { x : f ( x ) ≥ g ( x ) and g ( x ) = 0 } = [ k ( i ) P k ( i ) ∪ [ l ( j ) Q l ( j ) , where k ( i ) ∈ { , ..., m } and l ( j ) ∈ { , ..., n } . Then we notice Z S n − Ψ( u ) p dS p ( h f i p , u ) + Z S n − Ψ( u ) p dS p ( h g i p , u )= n ( Z R n Ψ( −∇ f ( x )) p dx + Z R n Ψ( −∇ g ( x )) p dx )= n ( m X i =1 Ψ( − u i ) p | P i | + j = n X j =1 Ψ( − v j ) p | Q j | ) . Similarly, Z S n − Ψ( u ) p dS p ( h f ∧ g i p , u ) + Z S n − Ψ( u ) p dS p ( h f ∨ g i p , u )= n ( Z R n Ψ( −∇ ( f ∧ g )( x )) p dx + Z R n Ψ( −∇ ( f ∧ g )( x )) p dx )= n ( X k ( i ) ,l ( j ) (Ψ( − u k ( i ) ) p | P k ( i ) | + Ψ( − v l ( j ) ) p | Q l ( j ) | )+ X i = k ( i ) Ψ( − u i ) p | P i | + X j = l ( j ) Ψ( − v j ) p | Q j | )= n ( m X i =1 Ψ( − u ) p | P i | + n X j =1 Ψ( − v j ) p | Q j | ) . This concludes the proof. (cid:3)
In [13], Monika Ludwig started the search of interesting valuationson function spaces. She found out that the only reasonable valuationthat enjoys some natural affine invariance property is the LYZ opera-tor defined by equation (1). Motivated by this, we propose a Ludwigtype conjecture: if Z : ( L , ( R n ) , ∧ , ∨ ) → ( P n , ♯ p ) is a continuous L p Blaschke valuation which satisfies Z ( f ) = Z ( f ◦ τ − ) , Z ( f ◦ φ − ) = | det φ | q φZ ( f )for all τ ∈ R n , φ ∈ GL ( n ) and some q ∈ R , then Z = c h·i p for some constant c ≥ . N THE DISCRETE FUNCTIONAL L p MINKOWSKI PROBLEM 11
The homothetic functional L p Minkowski problem.
Given f ∈ W ,p ( R n ), its distribution function µ f : [0 , ∞ ) → [0 , ∞ ] is definedby µ f ( t ) = |{ x ∈ R n : | f ( x ) | > t }| , where | · | denotes Lebesgue measure on R n . The decreasing rearrange-ment f ∗ : [0 , ∞ ) → [0 , ∞ ] of f is defined by f ∗ ( s ) = inf { t > µ f ( t ) ≤ s } . The symmetric rearrangement f ∗ : R n → [0 , ∞ ] of f is the functiondefined by f ⋆ ( x ) = f ∗ ( ω n | x | n ) , where | · | is the standard Euclidean norm.For a convex body K ∈ K n , the convex symmetrization f K of f withrespect to K is defined as follows: f K ( x ) = f ∗ ( ω n h ( ˜ K ◦ , x ) n ) , where h ( ˜ K ◦ , x ) is the support function of ˜ K ◦ , with ˜ K being a dilationof K so that | ˜ K | = ω n . Lemma 4.
Given f K ∈ W ,p ( R n ) and K ∈ K n , there is a uniqueconvex body h f K i p ∈ K n such that n Z R n Ψ p ( −∇ f K ( x )) dx = Z S n − Ψ( u ) p dS p ( h f K i p , u ) , for every continous Ψ that is homogeneous of degree . Furthermore h f K i p = ξ f ˜ K, where ξ f = ( n ( nω n ) p R ∞ t np + n − p − (( − f ∗ ) ′ ( ω n t n )) p dt ) n − p . Proof.
We remark that the proof here is similar to that in [29].First, we notice that the uniqueness is a quick consequence of the L p Minkowski inequality (Equation (1)).Since h ( ˜ K ◦ , · ) is a Lipschitz function and h ( ˜ K ◦ , · ) = 1 on ∂ ˜ K, foralmost all x ∈ ∂ ˜ K , σ ˜ K ( x ) = ∇ h ( ˜ K ◦ , x ) |∇ h ( ˜ K ◦ , x ) | , where ˜ σ K ( x ) is the outer unit normal vector of ˜ K at the point x . Andfrom the definition of the polar body, we have h ( ˜ K, σ K ( x )) = 1 |∇ h ( ˜ K ◦ , x ) | . Also, it is well known that f ∗ is locally absolutely continuous on(0 , ∞ ). We have, by the co-area formula applied to h ( ˜ K ◦ , · ), that: n Z R n Ψ p ( −∇ f K ( x )) dx = n Z R n Ψ p ( − ( f ∗ ) ′ ( ω n h ( ˜ K ◦ , x ) n ) nω n h ( ˜ K ◦ , x ) n − ∇ h ( ˜ K ◦ , x )) dx = n Z ∞ Z ∂K t n − (( − ( f ∗ ) ′ ( ω n t n )) nω n t n − ) p Ψ p ( ∇ h ( ˜ K ◦ , x ) |∇ h ( ˜ K ◦ , x ) | ) |∇ h ( ˜ K ◦ , x ) | p − dH n − ( x ) dt = n ( nω n ) p Z ∞ t np + n − p − (( − f ∗ ) ′ ( ω n t n )) p dt Z S n − Ψ p ( u ) h − p ( ˜ K, u ) dS ( ˜ K, u )= n ( nω n ) p Z ∞ t np + n − p − (( − f ∗ ) ′ ( ω n t n )) p dt Z S n − Ψ p ( u ) dS p ( ˜ K, u )for any Ψ that is homogeneous of degree 1.So we get S p ( h f K i p , u ) = ξ f S p ( ˜ K, u ) , where ξ f = n ( nω n ) p R ∞ t np + n − p − (( − f ∗ ) ′ ( ω n t n )) p dt. This concludes theproof. (cid:3)
Corollary 2. If f ∈ L , ( R n ) and K, L ∈ P n , then h f K i p is a dilate of K and |h f K i p | = |h f L i p | . Proof.
It is an immediate corollary of Lemma 4. Because from Lemma4, we have h f K i p = ξ f ˜ K, and h f L i p = ξ f ˜ L. We know from the definition that | ˜ K | = | ˜ L | , therefore Corollary 2follows. (cid:3) Applications
General L p Affine Sobolev Inequalities.
The general L p Sobolevinequality established in [1] could be stated as follows:
Theorem 8 ([1]) . Let K ∈ K n be normalized such that | K | = nω n .Then α n,p ( Z R n h ( K, ∇ f ( x )) p dx ) /p ≥ ( Z R n | f | p ⋆ dx ) /p ⋆ , N THE DISCRETE FUNCTIONAL L p MINKOWSKI PROBLEM 13 for < p < n and f ∈ W ,p ( R n ) , where p ⋆ = npn − p . The optimalconstants α n,p are given by (7) α n,p = n − /p ( p − n − p ) − /p ( Γ( n ) ω n Γ( np )Γ( n + 1 − np ) ) /n . By optimizing this inequality among all norms that have unit ballsin P n , we get the following result: Theorem 9.
For f ∈ L , ( R n ) with < p < n , we have ( ω n ) /n α n,p |h f i p | n − pnp ≥ ( Z R n | f | p ⋆ dx ) /p ⋆ , where the best constant α n,p is defined in Theorem 8 and p ⋆ = npn − p .Proof. Since we have α n,p ( Z R n h ( K, ∇ f ( x )) p dx ) /p ≥ ( Z R n | f | p ⋆ dx ) /p ⋆ , for arbitrary K ∈ K n with fixed volume nω n , we have, in particular, α n,p ( Z R n h ( ˜ h f i p , ∇ f ( x )) p dx ) /p ≥ ( Z R n | f | p ⋆ dx ) /p ⋆ . Therefore, we have α n,p |h f i p | n − pnp ≥ ( ω n ) − /n ( Z R n | f | p ⋆ dx ) /p ⋆ . (cid:3) Theorem 10.
Given ≤ λ ≤ and a function f ∈ W ,p ( R n ) with < p < n , we have /p α n,p ( Z S n − ( Z R n ((1 − λ )( D v f ) p + + λ ( D v f ) p − ) dx ) − n/p dv ) − /n ≥ ( Z R n | f | p ⋆ dx ) /p ⋆ , where the best constant α n,p is defined in Theorem 8 and p ⋆ = npn − p .Proof. For f ∈ L , ( R n ), since we have |h f i p | n − pnp ≥ ( ω n ) − /n α − n,p ( Z R n | f | p ⋆ dx ) /p ⋆ , by using Theorem 3 we have | Φ λ,p h f i p | − ≥ α |h f i p | n/p − , where Φ λ,p ( h f i p ) is defined through equation (4) as:Φ λ,p ( h f i p ) = (1 − λ ) · Π + p ( h f i p ) + p λ · Π − p ( h f i p ) , with 0 ≤ λ ≤ α is the best possible constant depending on p, n . Combining the above inequalities we get | Φ λ,p h f i p | − n ≥ α ( Z R n | f | p ⋆ dx ) /p ⋆ . Using the polar formula, we get( Z S n − ( Z R n ((1 − λ )( D v f ) p + + λ ( D v f ) p − ) dx ) − n/p dv ) − /n ≥ α ( Z R n | f | p ⋆ dx ) /p ⋆ , where α is the best possible constant depending on n, p. For general f ∈ W ,p ( R n ), we choose a sequence f k ∈ L , ( R n ) , suchthat f k → f in W ,p ( R n ). We know that Z R n ((1 − λ )( D v f k ) p + + λ ( D v f k ) p ) dx → Z R n ((1 − λ )( D v f ) p + + λ ( D v f ) p ) dx pointwise for all v ∈ S n − as k → ∞ .Since both ( R R n ((1 − λ )( D v f k ) p + + λ ( D v f k ) p − ) dx ) /p and ( R R n ((1 − λ )( D v f ) p + + λ ( D v f ) p − ) dx ) /p are support functions of convex bodies, weknow from Lemma 1 that Z R n ((1 − λ )( D v f k ) p + + λ ( D v f k ) p ) dx → Z R n ((1 − λ )( D v f ) p + + λ ( D v f ) p ) dx uniformly for all v ∈ S n − as k → ∞ . Therefore, using Fatou’s lemma we arrive at( Z S n − ( Z R n ((1 − λ )( D v f ) p + + λ ( D v f ) p − ) dx ) − n/p dv ) − /n ≥ α ( Z R n | f | p ⋆ ) /p ⋆ for all f ∈ W ,p ( R n ), where α is the best possible constant dependingon n, p. By calculating the optimal constant α , we conclude the proof. (cid:3) General Affine P´olya-Szeg¨o Principles.
While the usual P´olya-Szeg¨o principles states that the Dirichlet integral does not increase bysymmetric rearrangement. The affine P´olya-Szeg¨o principles states theaffine energy, which is affine invariant, defined in [22] does not increaseby symmetric rearrangement. Yet the affine P´olya-Szeg¨o principles isstronger than the usual P´olya-Szeg¨o principles in the sense that theaffine P´olya-Szeg¨o principles implies the usual P´olya-Szeg¨o principlesby the H¨older inequality.4.2.1.
The case p = n . We assume in this subsection that 1 < p < ∞ and p = n . We will deal with the case p = n in later parts because weneed a different normalization factor.In [1] the authors studied convex symmetrization for general norms.In terms of support functions of convex bodies, the result can be statedas follows: N THE DISCRETE FUNCTIONAL L p MINKOWSKI PROBLEM 15
Theorem 11 ([1]) . For K ∈ K n , Z R n h ( K, ∇ f ( x )) p dx ≥ Z R n h ( K, ∇ f K ( x )) p dx for all f ∈ L , ( R n ) . As an application of the above theorem, we have the following lemma:
Lemma 5.
Given f ∈ L , ( R n ) , we have h f i p = (1 + α ) h f h f i p i p for some α ≥ . Proof.
Since we know n Z R n h ( K, ∇ f ( x )) p dx = Z S n − h ( K, u ) p dS ( h f i p , u )= nV p ( h f i p , K ) . Similarly, we have n Z R n h ( K, ∇ f K ( x )) p dx = Z S n − h ( K, u ) p dS ( h f K i p , u )= nV p ( h f K i p , K ) . Since from Lemma 4 we know that h f K i p is a dilation of K , we have,by choosing K = h f i p in the above equation and using Theorem 11and Theorem 6, that V p ( h f i p , h f h f i p i p ) ≤ V p ( h f i p , h f i p ) . This implies that h f i p = (1 + α ) h f h f i p i p for some α ≥ . (cid:3) We are ready to prove the following general affine P´olya-Szeg¨o prin-ciple. We remark that the case λ = is due to Cianchi, Lutwak, Yangand Zhang [5] and the case λ = 0 is due to Haberl, Schuster and Xiao[9]. Theorem 12.
Given f ∈ W ,p ( R n ) , if we denote Ω λ,p ( f ) = α n,p ( Z S n − ( Z R n ((1 − λ )( D v f ) p + + λ ( D v f ) p − ) dx ) − n/p du ) − /n , where ≤ λ ≤ , α n,p = ( nω n ) /n ( nω n ω p − ω n + p ) /p and ω n = π n/ Γ(1+ n ) , wehave Ω λ,p ( f ) ≥ Ω λ,p ( f ⋆ ) . Proof. If f ∈ L , ( R n ), by Lemma 5, we have h f i p = (1 + α ) h f h f i p i p for some α ≥ . Therefore Φ λ,p ( h f i p ) ⊇ Φ λ,p ( h f h f i p i p ) , where Φ λ,p ( h f i p )is defined through Equation (4). It is not difficult to see that | Φ λ,p ( h f i p ) | ≤ | Φ λ,p ( h f h f i p i p ) | , therefore we have Ω λ,p ( f ) ≥ Ω λ,p ( f h f i p ) . On the other hand, by Lemma 4 and Theorem 3, we have | Φ λ,p ( h f h f i p i p ) | ≤ | Φ λ,p ( h f ⋆ i p ) | , which implies Ω λ,p ( f h f i p ) ≥ Ω λ,p ( f ⋆ ) . Therefore, we have Ω λ,p ( f ) ≥ Ω λ,p ( f ⋆ ) , for f ∈ L , ( R n ) . For general f ∈ W ,p ( R n ), consider an approximation sequence f k ∈ L , ( R n ) in the W ,p ( R n ) norm. We know thatΩ λ,p ( f ⋆k ) ≤ Ω λ,p ( f k ) f or k ∈ N. We know that Z R n ((1 − λ )( D v f k ) p + + λ ( D v f k ) p − ) dx → Z R n ((1 − λ )( D v f ) p + + λ ( D v f ) p − ) dx pointwise for all v ∈ S n − as k → ∞ .Since both ( R R n ((1 − λ )( D v f k ) p + λ ( D v f k ) p − ) dx ) /p and ( R R n ((1 − λ )( D v f ) p + + λ ( D v f ) p − ) dx ) /p are support functions of convex bodies, we know fromLemma 1 that Z R n ((1 − λ )( D v f k ) p + + λ ( D v f k ) p − ) dx → Z R n ((1 − λ )( D v f ) p + + λ ( D v f ) p − ) dx uniformly for all v ∈ S n − as k → ∞ . Moreover, the function v → ( Z R n ((1 − λ )( D v f ) p + + λ ( D v f ) p − ) dx ) /p is strictly positive and continuous on S n − , hence attains a positiveminimum on S n − . Consequently,( Z R n ((1 − λ )( D v f k ) p + + λ ( D v f k ) p ) dx ) − np → ( Z R n ((1 − λ )( D v f ) p + + λ ( D v f ) p − ) dx ) − np N THE DISCRETE FUNCTIONAL L p MINKOWSKI PROBLEM 17 uniformly for v ∈ S n − as k → ∞ . Thereforelim k →∞ Ω λ,p ( f k ) = Ω λ,p ( f ) . On the other hand, f ⋆ → f ⋆ in L p ( R n ), because of the contractiv-ity of the spherically symmetric rearrangement in L p ( R n ) . Hence weget that f ⋆k → f ⋆ weakly in W ,p ( R n ) . Since Ω λ,p ( f ⋆k ) = k∇ f ⋆k k p andΩ λ,p ( f ⋆ ) = k∇ f ⋆ k p , and since the L p ( R n ) norm of the gradient is lowersemi-continuous with respect to weak convergence in W ,p ( R n ) , lim inf k →∞ Ω λ,p ( f ⋆k ) ≥ Ω λ,p ( f ⋆ ) . Therefore, we have Ω λ,p ( f ⋆ ) ≤ Ω λ,p ( f ) , for all f ∈ W ,p ( R n ) . (cid:3) The case p = n . For dealing with the case p = n , we use thefollowing normalized version of the discrete functional L p Minkowskiproblem.
Theorem 13.
Given a function f ∈ L ,n ( R n ) , there exists a uniqueconvex polytope h f i n ∈ P n such that Z R n Ψ n ( −∇ f ( x )) dx = 1 |h f i n | Z S n − Ψ( u ) n dS n ( h f i n , u ) , for every continuous function Ψ : R n → [0 , ∞ ) that is homogeneous ofdegree . We remark that we omit the proof of Theorem 13, because it isessentially the same as the proof of Theorem 6. We remark that Lemma6 still holds. We will not give a proof of it. The proof is essentiallythe same as the proof of Lemma 5, if as we define the normalized L n optimal Sobolev body h f K i n as the convex body satisfying Z R n Ψ n ( −∇ f K ( x )) dx = 1 |h f K i n | Z S n − Ψ( u ) n dS n ( h f K i n , u ) , for every continuous function Ψ : R n → [0 , ∞ ) that is homogeneous ofdegree 1. Lemma 6.
Given f ∈ L , ( R n ) , we have h f h f i n i n = (1 + α ) h f i n for some α ≥ . Given 0 ≤ λ ≤
1, we define the normalized general L n projectionbody ˜Φ n K of K by: h n ( ˜Φ λ,n ( K ) , v ) = 1 | K | Z S n − ((1 − λ )( u · v ) n + + λ ( u · v ) n − ) dS n ( K, u ) . We need the following normalized version of the general L n Pettyprojection inequality.
Lemma 7 ([20]) . If K is a convex body, then | ˜Φ λ,n K || K | ≤ | ˜Φ λ,n B || B | , and equality holds if and only if K is an ellipsoid. With all these tools at hand, we establish the following theorem. Weremark again that the case λ = is due to Cianchi, Lutwak, Yang andZhang [5] and the case λ = 0 is due to Haberl, Schuster and Xiao [9]. Theorem 14.
Given f ∈ W ,n ( R n ) , we have Ω λ,n ( f ) ≥ Ω λ,n ( f ⋆ ) , if we denote Ω λ,n ( f ) = 2 /n α n ( Z S n − ( Z R n ((1 − λ )( D v f ) n + + λ ( D v f ) n − ) dx ) − dv ) − /n , where α n = ( nω n ) /n ( nω n ω n − ω n − ) /n . Proof.
By Lemma 6, we have h f h f i n i n ⊇ h f i n . Therefore, we have | ˜Φ ◦ λ,n h f h f i n i n | − n ≤ | ˜Φ ◦ λ,n h f i n | − n , for f ∈ L ,n ( R n ). By Lemma 7 we have | ˜Φ ◦ λ,n h f ⋆ i n | − n ≤ | ˜Φ ◦ λ,n h f h f i n i n | − n ≤ | ˜Φ ◦ λ,n h f i n | − n , which implies Ω λ,n ( f ) ≥ Ω λ,n ( f ⋆ ) f or all f ∈ L ,n ( R n ) . For general f ∈ W ,n ( R n ), the approximation argument is the sameas in Theorem 12. (cid:3) N THE DISCRETE FUNCTIONAL L p MINKOWSKI PROBLEM 19
General Affine Sobolev Type Inequalities.
For the symmet-ric case, the affine energy coincides with the Dirichlet integral:
Lemma 8.
Given f ∈ W ,p ( R n ) with < p < ∞ and ≤ λ ≤ , wehave Ω λ,p ( f ⋆ ) = k∇ f ⋆ k p . We omit the proof of Lemma 8 because it is essentially the same asthat in [9].With the general affne P´olya-Szeg¨o principle, Theorem 12 and The-orem 14 at hand, it is straightforward to get some affine Sobolev typeinequalities which generalize the results in [9]. We will list some (andnot all) of them here. Because the proofs are essentially the same asthose in [9], we will omit them.
Corollary 3 (General affine Moser-Trudinger inequalities) . If f ∈ W ,n ( R n ) with < | sprtf | < ∞ and ≤ λ ≤ , then (8) 1 | sprtf | Z sprtf exp( nω /nn | f ( x ) | Ω λ,n ( f ) ) nn − dx ≤ m n . Here sprtf is the support of f and the constant nω /nn is optimal, inthat equation (7) would fail for any real number m n if nω /nn were to bereplaced by a larger number. And the best constant m n is characterizedas m n = sup g Z ∞ exp( g ( t ) nn − − t ) dt, where the supremum ranges over all nondecreasing and locally ab-solutely continuous functions g on [0 , ∞ ) such that g (0) = 0 and R ∞ g ′ ( t ) n dt ≤ . Corollary 4 (General affine Morrey-Sobolev inequalities) . If f ∈ W ,p ( R n ) , p > n, such that | sprtf | < ∞ , then k f k ∞ ≤ α n,p | sprtf | p − nnp Ω λ,p ( f ) , where α n,p = n − /p ω − /nn ( p − p − n ) p − p is the best constant. Remark: Everything that is done in this paper has a counterpart forthe case p = 1. The extension is similar to what we’ve done in thispaper, as long as one observes Z R n ∇ f ( x ) dx = 0for every f ∈ W , ( R n ) . Acknowledgements
The work of the author was supported by Austrian Science Fund(FWF) Project P25515-N25.
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Institut f¨ur Diskrete Mathematik und Geometrie, TU Wien, Wied-ner Hauptstrasse 8-10, 1040 Wien, Austria
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