Interpolations, convexity and geometric inequalities
aa r X i v : . [ m a t h . F A ] S e p Interpolations, convexity and geometric inequalities
D. Cordero-Erausquin and B. Klartag Abstract
We survey some interplays between spectral estimates of H¨ormander-type, degener-ate Monge-Amp`ere equations and geometric inequalities related to log-concavity such asBrunn-Minkowski, Santal´o or Busemann inequalities.
The Brunn-Minkowski inequality has an L interpretation, an observation that can be tracedback to the proof provided by Hilbert. More recently, it has been noted that the Brunn-Minkowski inequality for convex bodies is related, in its local form, to spectral inequalities. Infact, the Pr´ekopa theorem, which is the function form of the Brunn-Minkowski inequality forconvex sets, is equivalent to spectral inequalities of Brascam-Lieb type. The local derivationof Pr´ekopa’s theorem from spectral L inequalities was described in the more general complexsetting in [13] and then extended further in [6, 7].Let K , K ⊂ R n be two convex bodies (i.e., compact convex sets with non-empty interior)and denote, for t ∈ [0 , K ( t ) := (1 − t ) K + tK = { z ∈ R n ; ∃ ( a, b ) ∈ K × K , z = (1 − t ) a + tb } . (1)The Brunn-Minkowski inequality is central in the theory of convex bodies. Denoting theLebesgue measure by | · | , it states that | K ( t ) | ≥ | K | − t | K | t , with equality if and only if K = K + x for x ∈ R n . Introducing the convex body K := [ t ∈ [0 , { t } × K ( t ) ⊂ R n +1 , then K ( t ) is the section over t , and the Brunn-Minkowski inequality expresses the log-concavity of the marginal measure. Namely, it shows that the function α ( t ) := − log | K ( t ) | is convex. The Brunn-Minkowski inequality for convex bodies admits the following usefulfunctional form, which states that marginals of log-concave functions are log-concave. Institut de Math´ematiques de Jussieu, Universit´e Pierre et Marie Curie (Paris 6), 4 place Jussieu, 75252Paris, France. Email: [email protected]. School of Mathematical Sciences, Tel-Aviv University, Tel Aviv 69978, Israel. Supported in part by theIsrael Science Foundation and by a Marie Curie Reintegration Grant from the Commission of the EuropeanCommunities. Email: [email protected] heorem 1 (Pr´ekopa) . Let F : R n +1 → R ∪ { + ∞} be convex with R exp( − F ) < ∞ anddefine α : R −→ R ∪ { + ∞} by e − α ( t ) = Z R n e − F ( t,x ) dx. Then α is convex. The Brunn-Minkowski inequality then follows by considering, for a given convex set K ⊂ R n +1 = R × R n , the convex function F defined by e − F ( t,x ) = K ( t, x ) = K ( t ) ( x ) . (2)The standard proofs of Brunn-Minkowski rely on parameterization or mass transporttechniques between K and K , with the parameter t ∈ [0 ,
1] being fixed. A natural questionis whether one can provide a direct local approach by proving α ′′ ( t ) ≥
0? The answer isaffirmative and this was shown recently by Ball, Barthe and Naor [4]. As mentioned earlier,this local approach was put forward in an L framework, for analogous complex versions, inCordero-Erausquin [13] and in subsequent far-reaching works by Berndtsson [6, 7].Another essential concept in the theory of convex bodies is duality. This requires us tofix a center and a scalar product. Let x · y stand for the standard scalar product of x, y ∈ R n .We write | x | = x · x and B n = { x ∈ R n ; x · x ≤ } , the associated unit ball. Recall that K ⊂ R n is a centrally-symmetric convex body if and only if K is the unit ball for some norm k · k on R n , a relation denoted by K = B k·k := { x ∈ R n ; k x k ≤ } . The polar of K is definedas the unit ball of the dual norm k · k ∗ , K ◦ = B k·k ∗ = { y ∈ R n ; x · y ≤ , ∀ x ∈ K } . We have the following beautiful result:
Theorem 2 (Blaschke-Santal´o inequality) . For every centrally-symmetric convex body K ⊂ R n , we have | K | | K ◦ | ≤ | B n | with equality holding true if and only if K is an ellipsoid (i.e. a linear image of B n ). The corresponding functional form reads as follows (see [1, 2]): for an even function f : R n → R with 0 < R e − f < ∞ , if L f denotes its Legendre transform, then Z e − f Z e −L f ≤ (cid:16) Z e −| x | / dx (cid:17) = (2 π ) n . (3)Note that the Brunn-Minkowski inequality entails p | K | | K ◦ | ≤ (cid:12)(cid:12)(cid:12)(cid:12) K + K ◦ (cid:12)(cid:12)(cid:12)(cid:12) . However, in general we have K + K ◦ ! B n . For instance, take K = T ( B n ), where T = Id R n is a positive-definite symmetric operator. Then K ◦ = T − ( B n ). Observe that K + K ◦ ⊃ T + T − ( B n ) and T + T − > √ T T − = Id R n
2n the sense of symmetric matrices. This suggest that instead of taking convex combinations,as in the Brunn-Minkowski theory, we would like to consider geometric means of convexbodies. It turns out that this is exactly what complex interpolation does, and it is a challengingquestion to understand real analogues of this procedure.In this note we will consider several ways of going from K to K , or equivalently from anorm k · k to another norm k · k . There are many ways to recover the volume of K fromthe associated norm k · k . Let p > n ≥
1. There exists an explicit constant c n,p > K ⊂ R n , with associated norm k · k K ,we have Z R n e −k x k pK /p dx = c n,p | K | . (4)Note that the procedure (2) corresponds to the case p → + ∞ .We aim to find ways of interpolating between norms in order to recover, among otherthings, the Brunn-Minkowski and the Santal´o inequalities.Let us next put forward some notation as well as a formula that we shall use throughoutthe paper. Notation 3.
For a function F : R n → R such that R e − F ( x ) dx < + ∞ , we denote by µ F theprobability measure on R n given by dµ F ( x ) := e − F ( x ) R e − F dx. For a function of n + 1 variables F : I × R n → R , where I is an interval of R , we denote,for a fixed t ∈ I , F t := F ( t, · ) : R n → R and then by µ F t the corresponding probability measureon R n . We also set α ( t ) = − log Z R n e − F t ( x ) dx. The variance with respect to a probability measure µ of a function u ∈ L ( µ ) – where,depending on the context, we consider either real-valued or complex-valued functions – isdefined as the L norm of the projection of u onto the space of functions orthogonal toconstant functions, i.e.Var µ ( u ) := Z (cid:12)(cid:12) u − R u dµ (cid:12)(cid:12) dµ = Z | u | dµ − (cid:12)(cid:12)(cid:12) Z u dµ (cid:12)(cid:12)(cid:12) . A straightforward computation yields:
Fact 4.
With Notation 3, we have for every t ∈ I , α ′′ ( t ) = Z R n ∂ tt F dµ F t ( x ) − "Z R n (cid:0) ∂ t F ( t, x ) (cid:1) dµ F t ( x ) − (cid:18)Z R n ∂ t F ( t, x ) dµ F t ( x ) (cid:19) = Z R n ∂ tt F dµ F t − Var µ Ft (cid:0) ∂ t F (cid:1) , (5) assuming that F is sufficiently regular to allow for the differentiations under the integral sign. F the function α is convex, bylooking at α ′′ . Actually, we will first discuss the complex case, where convexity is replaced byplurisubharmonicity. We will recover the fact that families given by complex interpolation,or equivalently by degenerate Monge-Ampre equations, lead to subharmonic functions α .Then we will try to see, at a very heuristic level, what can be said in the real case. A finalsection proposes a local L approach, to the Busemann inequality, similar to that used in thepreceding sections. Acknowledgement.
We thank Yanir Rubinstein and Bo Berndtsson for interesting, relateddiscussions.
Let K and K be two unit balls of C n associated with the (complex vector space) norms k · k and k · k . Note that here we are working with the class of convex bodies K of R n thatare circled , meaning that e iθ K = K for every θ ∈ R . We think of a normed space as a tripletconsisting of a vector space, a norm and its unit ball. Consider the complex normed spaces X = ( C n , k · k , K ) and X = ( C n , k · k , K ) and write X z = ( C n , k · k z , K z )for the complex Calder´on interpolated space at z ∈ C := { w ∈ C ; ℜ ( w ) ∈ [0 , } where ℜ ( w ) is the real part of w ∈ C . Recall that X z = X ℜ ( z ) and therefore K z = K t with t = ℜ ( z ) ∈ [0 , Theorem 5 ([12]) . The function t → | K t | is log-concave on [0 , and so | K | − t | K | t ≤ | K t | . (6)In the case of complex unit balls, this result improves upon the Brunn-Minkowski inequal-ity since it can be verified, by using the Poisson kernel on [0 , × C n and the definition of theinterpolated norm, that K t ⊂ (1 − t ) K + tK = K ( t ) . In this setting, it also gives the Santal´o inequality. Indeed, for a given complex unit ball K ⊂ C n , let X be the associated complex normed space, and let X be the dual conjugatespace which has K ◦ ⊂ C n as its unit ball. Then it is well known that X / = ℓ n ( C ) = ℓ n ( R ) (7)and therefore we obtain p | K | | K ◦ | ≤ | B n | . (Let us mention here that the conjugation bar in the statements of [12] is superfluous accordingto standard definitions). 4n order to have a better grasp on complex interpolation, let us write an explicit formulain the specific case of Reinhardt domains. A subset K ⊂ C n is Reinhardt if for any z =( z , . . . , z n ) ∈ C n , ( z , . . . , z n ) ∈ K ⇔ ( | z | , . . . , | z n | ) ∈ K. Note that a Reinhardt convex set is necessarily circled. In the case where X = ( C n , k · k , K )and X = ( C n , k · k , K ) are such that K and K are Reinhardt, the interpolated space X z = ( C n , k · k z , K z ) satisfies K z = (cid:8) z ∈ C n ; ∃ ( a, b ) ∈ K × K , | z j | = | a j | − t | b j | t for j = 1 , . . . , n (cid:9) with t = ℜ ( z ). The case of Reinhardt unit balls is particularly simple and easy to analyze,but it has its limitations. Still, the idea is that in general, K t should be understood as a“geometric mean” of the bodies K and K , whereas the Minkowski sum (1) reminds us ofan arithmetic mean.Theorem 5 was proved using the complex version of the Pr´ekopa theorem obtained byBerndtsson [5], which was derived in [13] using a local computation and L spectral inequalitiesof H¨ordmander type. Here, we would like to provide a different direct proof, by combiningthe results of Rochberg and H¨ormander’s a priori L -estimates. Let k · k z be a family ofinterpolated norms on C n and K z = B k·k z . We assume for simplicity that these normsare smooth and strictly convex, so that we will not have to worry about justification ofthe differentiations under the integral signs. In fact, by approximation we can assume that1 /R ≤ Hess k · k k ≤ R (for some large constant R >
1) for k = 1 ,
2, and these bounds remainvalid for the interpolated norms. Introduce the function F : C × C n → R , F ( z, w ) := 12 k w k z . Denote the Lebesgue measure on C n ≃ R n by λ , and introduce, in view of (4), α ( z ) = − log Z C n e − F ( z,w ) dλ ( w ) = − log | K z | − log( c n, )for z ∈ C . Our goal is to prove that t → α ( t ) is convex on [0 , α ( z ) = α ( ℜ ( z )), thisis equivalent to proving that α is subharmonic on the strip C . The following analogue of (5)is also straightforward:14 ∆ α ( z ) = ∂ zz α ( z ) = Z C n ∂ zz F dµ F z − Z C n (cid:12)(cid:12) ∂ z F ( w ) − R ∂ z F dµ F z (cid:12)(cid:12) dµ F z ( w ) , where µ F z is the probability measure on C n given by dµ F z ( w ) = e − F ( z,w ) R e − F ( z,ζ ) dλ ( ζ ) dλ ( w ).It was explained by Rochberg [17] that complex interpolation is characterized by thefollowing differential equation: ∂ zz F = n X j,k =1 F jk ( z, w ) ∂ w j ( ∂ z F ) ∂ w k ( ∂ z F ) (8)where ( F jk ) j,k ≤ n is the inverse of the complex Hessian in the w -variables of F ( z, w ), that is (cid:16) F jk (cid:17) j,k ≤ n = (cid:16) Hess C w F (cid:17) − := (cid:20)(cid:16) ∂ w j w k F (cid:17) j,k ≤ n (cid:21) − . F is plurisubharmonic on C × C n ⊂ C n +1 and (8) expresses the factthat it is a solution of the degenerate Monge-Amp`ere equationdet (cid:16) Hess C z,w F (cid:17) = 0where Hess C z,w F is the full complex Hessian of F ( z, w ), an ( n + 1) × ( n + 1) matrix.As a consequence of the previous discussion, we have that, for a fixed z ∈ C and setting u := ∂ z F ( z, · ) : C n → C ,∆ α ( z ) / Z C n n X j,k =1 F jk ∂ w j u ∂ w k u dµ F z − Z (cid:12)(cid:12) u − R u dµ F z (cid:12)(cid:12) dµ F z . (9)Of course, it is now irresistible to appeal to H¨ormander’s a priori estimate (see e.g. [15]). Itstates that if F : C n → R is a (strictly) plurisubharmonic function and if u is a (smoothenough) function, then Z C n | u − P H u | dµ F ≤ Z C n n X j,k =1 F jk ∂ w j u ∂ w k u dµ F (10)where dµ F ( w ) = e − F ( w ) R e − F dλ dλ ( w ) and P H : L ( µ F ) → L ( µ F ) is the orthogonal projectiononto the closed space H = { h ∈ L ( µ F ) ; ∂h = 0 } of holomorphic functions. Actually, this a priori estimate on C n is rather easy to prove by duality and integration by parts. We nowapply this result to F = F ( z, · ), µ F = µ F z and u = ∂ z F . Note that F (and thus µ F ) and u are invariant under the action of S : F ( z, e iθ w ) = F ( z, w ) and the same is true for ∂ z F .This implies that the function P H u has the same invariance, but since it is a holomorphicfunction on C n , it has to be constant. Therefore P H u = R udµ F z and we indeed obtain that∆ α ( z ) ≥ R n . However, the real case is more complex, as we shall now see. The concept of interpolation and the basic properties we present here are due to Semmes [18],building on previous work by Rochberg [17]. Semmes indeed raised the question of whethersuch interpolations (which are not interpolations in the operator sense) could be used toprove inequalities, by showing that certain functionals are convex along the interpolation.6ur main contribution here is to explain that this is indeed the case, by connecting thisinterpolation with some well-known spectral inequalities. However, some discussions willremain at a heuristic level, as it is not the purpose of this note to discuss existence, unicityand regularity of solutions to the partial differential equations we refer to.
Definition 1 (Rochberg-Semmes interpolation [18]) . Let I be an interval of R and p ∈ [1 , + ∞ ] . We say that a smooth function F : I × R n → R is a family of p -interpolation if forany t ∈ I , the function F ( t, · ) is (strongly) convex on R n and for ( t, x ) ∈ I × R n ∂ tt F = 1 p (cid:0) Hess x F (cid:1) − ∇ ∂ t F · ∇ ∂ t F. (11) Accordingly, when ∂ tt F ≥ p (cid:0) Hess x F (cid:1) − ∇ ∂ t F · ∇ ∂ t F , we say that F is a sub-family of p -interpolation. In Definition 1, we denote by ∇ F the gradient of F ( t, x ) in the x variables, and a functionis strongly convex when Hess x F >
0. By standard linear algebra we have the followingequivalent formulation in terms of the degenerate Monge-Amp`ere equation:
Proposition 6 (Interpolation and degenerate Monge-Amp`ere equation) . Let F : I × R n → R be a smooth function such that F ( t, · ) is (strongly) convex on R n and introduce, for ( t, x ) ∈ I × R n , the ( n + 1) × ( n + 1) matrix H = H p F ( t, x ) := ∂ tt F ( ∇ x ∂ t F ) ∗ ∇ x ∂ t F p
Hess x F . (12) Then, F is a family (resp. a sub-family) of p -interpolation if and only if det H = 0 (resp. det H ≥ ) on I × R n . In particular, 1-interpolation corresponds exactly to the degenerate Monge-Amp`ere equa-tion on I × R n . In fact, we see p -interpolation as a (Dirichlet) boundary value problem. Definition 2.
Let F and F be two smooth convex functions on R n . We say that { F t : R n → R } ∈ [0 , is a p -interpolated family associated with { F , F } if F ( t, x ) = F t ( x ) is a family of p -interpolation on [0 , × R n with boundary value F (0 , · ) = F and F (1 , · ) = F . As we said above, we will not discuss in this exposition questions related to existence,uniqueness and regularity of solutions to this Dirichlet problem (except for the easy case p = 1, explained below). However, it is reasonable to expect that generalized solutions, whichare sufficient for our purposes, can be constructed by using Perron processes, as mentionedby Semmes [18].Using Notation 3, given a family or a sub-family of p -interpolation F , we aim to understandthe convexity of the function on I , α ( t ) = − log Z R n e − F ( t,x ) dx. (13)In view of (5), we see that for every fixed t ∈ I we have the implicationVar µ Ft ( ∂ t F ) ≤ p Z R n (cid:0) Hess x F (cid:1) − ∇ ∂ t F · ∇ ∂ t F dµ F t = ⇒ α ′′ ( t ) ≥ , (14)7nder some mild regularity assumptions. The left-hand side is of course reminiscent of thereal version of H¨ormander’s estimate (10), which is known as the Brascamp-Lieb from [9].Recall that this inequality states that if F : R n → R is a (strongly) convex function and if u ∈ L ( µ F ) is a locally Lipschitz function, thenVar µ F ( u ) ≤ Z R n (cid:0) Hess x F (cid:1) − ∇ u · ∇ u dµ F , (15)with our notation dµ F ( x ) = e − F ( x ) R e − F dx . Again, this inequality can easily be proven along thelines of H¨ormander’s approach (see below).Applying the Brascamp-Lieb inequality (15) to F = F ( t, · ) and u = ∂ t F when F is a1-interpolation sub-family, we obtain, in view of (14), the following statement: Proposition 7. If F is a sub-family of -interpolation, then α is convex. The first comment is that we have not proved anything new! Indeed, it is directly verifiedbelow that for any C -smooth function F , F is a sub-family of 1-interpolation ⇐⇒ F is convex on I × R n . (16)Therefore, we have reproduced Pr´ekopa’s Theorem 1. In order to demonstrate (16), observethat the positive semi-definiteness of the matrix H F ( t, x ) amounts to the inequality(Hess x F ) y · y + 2 ∇ x ( ∂ t F ) · y + ∂ tt F ≥ y ∈ R n , or equivalently, ∂ tt F ≥ sup y ∈ R n [2 ∇ x ( ∂ t F ) · y − (Hess x ) F y · y ] = (cid:0) Hess x F (cid:1) − ∇ x ∂ t F · ∇ x ∂ t F, as Hess x F is positive definite. Let us note that if F and F are given, then the associatedfamily of 1-interpolation – equivalently, the unique solution to the degenerate Monge-Amp`ereequation on [0 , × R n with F ( t, x ) convex in x – is F ( t, w ) = inf w =(1 − t ) x + ty (cid:8) (1 − t ) F ( x ) + tF ( y ) (cid:9) . (17)Every sub-family of 1-interpolation is above this F , and thus the statement of Pr´ekopa’sTheorem reduces to 1-interpolation families (an argument that is standard in the study offunctional Brunn-Minkowski inequalities). One way to recover the Brunn-Minkowski inequal-ity directly from this family F of 1-interpolation, is to take, as in the derivation from Pr´ekopa’stheorem, something like F ( x ) = k x k qK /q , F ( y ) := k y k qK /q and let q → + ∞ .We have just shown that Pr´ekopa’s theorem reduces, locally, to the Brascamp-Lieb in-equality. This is parallel to the complex setting, i.e to the local L -proof of the complexPr´ekopa theorem of Berndtsson given in [13] and extended in [6, 7]. The converse proce-dure was known, starting from the work of Brascamp and Lieb; more explicitely, Bobkov andLedoux [8] noted that the Pr´ekopa-Leindler inequality (an extension of Pr´ekopa’s result tothe case fibers are not convex) indeed implies the Brascamp-Lieb inequality. We also em-phasize Colesanti’s work [11], where, starting from the Brunn-Minkowski inequality, spectral8nequalities of Brascamp-Lieb type on the boundary ∂K of a convex body K ⊂ R n are ob-tained. This can also be recovered by applying the Brascamp-Lieb inequality to homogeneousfunctions. The conclusion is that all of these results are the global/local versions of the samephenomena. At the local level, we have reduced the problem to the inequality (15) whichexpresses a spectral bound in L ( µ F ) for the elliptic operator associated with the Dirichletform on the right-hand side of (15).For completeness, we would like to briefly recall here H¨ormander’s original approachto (15). Consider the Laplace-type operator on L ( µ F ), L := ∆ − ∇ F · ∇ , that we define, say, on C -smooth compactly supported functions. First, recall the integrationby parts formulae, R uLϕ dµ F = − R ∇ u · ∇ ϕ dµ F and Z R n ( Lϕ ) dµ F = Z R n (Hess x F ) ∇ ϕ · ∇ ϕ dµ F + Z R n k Hess x ϕ k dµ F , (18)where k Hess ϕ k = P i,j ≤ n ( ∂ i,j ϕ ) . Let u be a locally-Lipschitz function on R n . We usethe (rather weak) standard observation that the image by L of the C -smooth compactlysupported functions is dense in the space of L ( µ F ) functions orthogonal to constants (seee.g. [14]). For ε > ϕ be a C -smooth, compactly-supported function such that Lϕ − ( u − R udµ F ) has L ( µ F )-norm smaller than ε . Then, by integration by parts and using (18)we getVar µ F ( u ) = 2 Z (cid:0) u − R u dµ F (cid:1) Lϕ dµ F − Z ( Lϕ ) dµ F + Z (cid:0) Lϕ − (cid:0) u − R u dµ F (cid:1)(cid:1) dµ F ≤ − Z ∇ u · ∇ ϕ dµ F − Z (Hess x F ) ∇ ϕ · ∇ ϕ dµ F − Z k Hess x ϕ k dµ F + ε ≤ − Z ∇ u · ∇ ϕ − Z (Hess x F ) ∇ ϕ · ∇ ϕ dµ F + ε ≤ Z (cid:0)
Hess x F (cid:1) − ∇ u · ∇ u dµ F + ε , and (15) follows by letting ε tend to zero.Let us go back to interpolation families. As we said, 1-sub-interpolation corresponds to afunction F that is convex on I × R n . More generally, we have the following characterization,proved by Semmes: Proposition 8.
For a smooth function F : I × R n → R , the following are equivalent: • F is a sub-family of p -interpolation. • With the notation (12) , we have, ∀ ( t, x ) ∈ I × R n , H p F ( t, x ) ≥ . • For all x , y ∈ R n , the function ( s, t ) −→ F (cid:16) t, x + ( t + p p − s ) y (cid:17) is subharmonic on the subset of R where it is defined. F of p -interpolation even when F is not very smooth.We turn now to duality, which was part of the motivation of Semmes. We shall denote by L the Legendre transform in space, i.e. on R n . In particular, for F : I × R n , we shall write L F ( t, x ) = L ( F t )( x ) = sup y ∈ R n (cid:8) x · y − F ( t, y ) (cid:9) . It is classical that if F is the family of 1-interpolation given by (17), then L F is a family of ∞ -interpolation, meaning that L F is affine in t : L F t ( x ) = (1 − t ) L F ( x ) + t L F ( x ) . So in this case, when we move to the dual setting, Brunn-Minkowski or Pr´ekopa’s inequalityis replaced by the trivial fact that α ( t ) = − log R e −L t F ( x ) dx is concave by H¨older’s inequality.More general duality relations hold for p -interpolations. Suppose F ( t, x ) = F t ( x ) is convexin x , and denote G ( t, y ) = L F t ( y ). We have the identity (proved below): ∂ tt F + ∂ tt G = (Hess x F ) − ∇ ∂ t F · ∇ ∂ t F = (Hess y G ) − ∇ ∂ t G · ∇ ∂ t G, (19)where F and its derivatives are evaluated at ( t, x ), while G and its derivatives are evaluatedat ( t, y ) = ( t, ∇ F ( x )). From this identity, we immediately conclude Proposition 9. If F is a family of p -interpolation, then L F is a family of p ′ -interpolation,where p ′ + p = 1 . We now present the details of the straightforward proof of (19). From the definition, G ( t, ∇ F ( t, x )) = h x, ∇ F ( t, x ) i − F ( t, x ) , (20) ∇ G t ( ∇ F t ( x )) = ( ∇ G )( t, ∇ F ( x )) = x (21)Hess y G ( t, ∇ F ( x, t )) = (Hess x F ( t, x )) − . (22)where the gradients and the hessians refer only to the space variables x, y . By differentiating(21) with respect to t , we see that ∇ ∂ t G = − (Hess y G )( ∇ ∂ t F ) (23)where G and its derivatives are evaluated at ( t, y ) = ( t, ∇ F ( x )), while F and its derivativesare evaluated at ( t, x ). From (22) and (23), − ∇ ∂ t G · ∇ ∂ t F = (Hess x F ) − ∇ ∂ t F · ∇ ∂ t F = (Hess y G ) − ∇ ∂ t G · ∇ ∂ t G. (24)Differentiating (20) with respect to t and using (21) we get that ∂ t G ( t, ∇ F ( x )) = − ∂ t F ( t, x ).If we differentiate this last equality one more time with respect to t , we find ∂ tt G + ∇ ∂ t G · ∇ ∂ t F = − ∂ tt F, which combined with (24) yields the desired formula (19).As a consequence of Proposition 8, we see that 2-interpolation families satisfy an interpo-lation duality theorem. Let f be a convex function on R n , and suppose that F t ( x ) = F ( t, x )is the 2-interpolation family F with F = f and F = L f . Then, F ( t, x ) = L F (1 − t, x )10rovided we have unicity for the 2-interpolation problem, and therefore we have F (cid:18) , x (cid:19) = | x | . If we take f ( x ) = k x k K /
2, then L f ( x ) = k x k K ◦ /
2. Thus, if we could prove that fora 2-interpolation family F , the associated function α from (13) is convex, as it is for 1-interpolations, then we would recover Santal´o’s inequality. This would be the case if we hada Brascamp-Lieb inequality with a factor / on the right-hand side of (15) for every convexfunction F : R n → R . However, this is of course false in general. Recall that even for the San-tal´o inequaliy, some “center” must be fixed or some symmetry must be assumed. Therefore,a more reasonable question to ask, is whether α is convex when the initial data f is even.This guarantees that F t is even for all t ∈ [0 , / in the right-hand side of (15) when F and u areeven, as can be shown by taking a perturbation of the Gaussian measure. This suggests thatthe answer to the question could be negative in general. A reasonable conjecture, perhaps,is: Conjecture 10.
Assume F and F are even, convex and -homogeneous (i.e. F i ( x ) = λ i k x k K i for some centrally-symmetric convex bodies K i ⊂ R n ), properties that propagatealong the interpolation. Then, the function α associated with the -interpolation family isconvex. Here is a much more modest result:
Fact 11.
Assume that f is convex and even, and let F be a -interpolation family with F = f and F = L f , with the associated function α as in (13) . Then, one has α ′′ (1 / ≥ . Proof.
Since F ( , x ) = | x | /
2, the probability measure µ F / is exactly the Gaussian measureon R n , which we denote by γ . Note also that Hess x F / = Id R n . Therefore, if we denote u = ∂ t F ( , · ), we need to check thatVar γ ( u ) ≤ Z R n |∇ u | dγ. The function v := u − R u dγ is by construction orthogonal to constant functions in L ( γ ). Butsince u is even (because F t is even for all t , and so is ∂ t F ), this function v is also orthogonalto linear functions. Recall that the Hermite (or Ornstein-Uhlenbeck) operator L = ∆ − x · ∇ has non-positive integers as eigenvalues, and that the eigenspaces (generated by Hermitepolynomials) associated with the eigenvalues 0 and − v belongs to the subspace where − L ≥ γ ( u ) = Z | v | dγ ≤ − Z vLv dγ = 12 Z |∇ u | dγ.
11e conclude this section by mentioning that we have analogous formulas in the case wherewe work with some fixed measure ν on R n , in place of the Lebesgue measure. Then, for afunction F : R n → R such that R e − F dν < + ∞ , we denote by µ ν,F the probability measureon R n given by dµ ν,F ( x ) := e − F ( x ) R e − F dν dν ( x ) . For a function of n + 1 variables F : I × R n → R , we denote as before F t := F ( t, · ) : R n → R and then µ ν,F t is the corresponding probability measure on R n . We are then interested in theconvexity of the function α ν ( t ) := − log Z R n e − F ( t,x ) dν ( x ) = − log Z R n e − F t dν. The computation is identical: α ′′ ν ( t ) = Z R n ∂ tt F dµ ν,F t − Var µ ν,Ft (cid:0) ∂ t F (cid:1) . Here is an illustration. Let ν be a symmetric log-concave measure on R n : dν ( x ) = e − W ( w ) dx with W being convex and even on R n , and consider the family F ( t, x ) = e t | x | / . This is a typical example of a 2-interpolation family. Then, the fact that the corresponding α ν is convex is equivalent to the B -conjecture proved in [14]. The argument there beginswith the computation above. It turns out that for this particular family F , the requiredBrascamp-Lieb inequality reduces to a Poincar´e inequality for the measure µ ν,F t , which holdsprecisely with a constant 1 / R n , as noted in Klartag [16]. Several examples of this type suggest that theLebesgue measure can often be replaced by a more general log-concave measure. We conclude this survey with a proof of the Busemann inequality via L inequalities. TheBusemann inequality [10] is concerned with non-parallel hyperplane sections of a convex body K ⊂ R n . In the particular case where K is centrally-symmetric, the Busemann inequalitystates that g ( x ) = | x || K ∩ x ⊥ | ( x ∈ R n )is a norm on R n . Here | K ∩ x ⊥ | is the ( n − K ∩ x ⊥ = { y ∈ K ; y · x = 0 } , and g (0) = 0 as interpreted by continuity. The convexity ofthe function g is a non-trivial fact. Using the Brunn-Minkowski inequality, the convexity of g reduces to a statement about log-concave functions in the plane, as observed by Busemann.Indeed, the convexity of g has to be checked along affine lines, and therefore on 2-dimensional12ector subspaces. Specifically, let E ⊂ R n be a two-dimensional plane, which we convenientlyidentify with R . For y ∈ R = E set e − w ( y ) = | K ∩ ( y + E ⊥ ) | , the ( n − K . Then w : R → R ∪ { + ∞} is a convexfunction, according to the Brunn-Minkowski inequality. For p > t ∈ R define α p ( t ) = Z ∞ e − w ( ts,s ) s p − ds. (25)Note that when K is centrally-symmetric, 2 √ t α ( t ) = | K ∩ (1 , − t ) ⊥ | . We therefore seethat Busemann’s inequality amounts to the convexity of the function 1 /α ( t ) on R . Next wewill prove the following more general statement, which is due to Ball [3] when p ≥ Theorem 12.
Let X be an n -dimensional real linear space and let w : X → R be a convexfunction with R e − w < ∞ . For p > and = x ∈ X denote h ( x ) = (cid:18)Z ∞ e − w ( sx ) s p − ds (cid:19) − /p with h (0) = 0 . Then h is a convex function on X . Busemann’s proof of the case p = 1 of Theorem 12, and the generalization to p ≥ Proof of Theorem 12:
By a standard approximation argument, we may assume that w issmooth and 1 /R ≤ Hess( w ) ≤ R at all points of R n , for some large constant R >
1. Therefore h is a continuous function, smooth outside the origin, and homogeneous of degree one. Sinceconvexity of a function involves three collinear points contained in a two-dimensional subspace,we may assume that n = 2. Thus, selecting a point 0 = z ∈ X and a direction θ ∈ X , our goalis to show that ∂ θθ h ( z ) ≥ h is homogeneous of degree one, it suffices to consider thecase z = 0). If θ is proportional to z , then the second derivative vanishes as h is homogeneousof degree one. We may therefore select coordinates ( t, x ) ∈ R = X , and identify z = (0 , θ = (1 , (cid:16) α − /pp (cid:17) ′′ (0) ≥ , where α p is defined in (25). Equivalently, we need to prove that at the origin, ∂ tt α p ≤ (cid:18) p (cid:19) ( ∂ t α p ) /α p . (26)We denote by µ the probability measure on [0 , ∞ ) whose density is proportional to theintegrable function exp( − w (0 , x )) x p − . Similarly to Fact 5 above with F ( t, x ) = w ( tx, x ), thedesired inequality (26) is equivalent toVar µ ( x∂ t w ) ≤ Z ∞ x ( ∂ tt w ) dµ ( x ) + 1 p (cid:18)Z ∞ x ( ∂ t w ) dµ ( x ) (cid:19) . (27)13e will use the convexity of w ( t, x ) via the inequality ∂ tt w ≥ (cid:0) ∂ tx w (cid:1) /∂ xx w , which expressesthe fact that w t ( x ) = w ( t, x ) is a sub-family of 1-interpolation. Denote u ( x ) = x∂ t w (0 , x )and compute that x∂ tx w = u ′ − u ( x ) /x for x >
0. Hence, in order to prove (27), it suffices toshow that Var µ ( u ) ≤ Z ∞ ∂ xx w (cid:18) u ′ ( x ) − u ( x ) x (cid:19) dµ ( x ) + 1 p (cid:18)Z ∞ udµ ( x ) (cid:19) . (28)We will prove (28) for any smooth function u ∈ L ( µ ) (it is clear that the function x∂ t w (0 , x )grows at most polynomially at infinity, and hence belongs to L ( µ )). By approximation,it suffices to restrict our attention to smooth functions such that u − R udµ is compactly-supported in [0 , ∞ ). Consider the Laplace-type operator Lϕ = ϕ ′′ − (cid:16) ∂ x w (0 , x ) − p − x (cid:17) ϕ ′ = ϕ ′′ − ∂ x (cid:16) w (0 , x ) − ( p −
1) log( x ) (cid:17) ϕ ′ . Integrating the ordinary differential equation, we find a smooth function ϕ , with ϕ ′ (0) = 0and ϕ ′ compactly-supported in [0 , ∞ ), such that Lϕ = u − R udµ . As before, we have theintegration by parts R ( Lϕ ) u dµ = − R ϕ ′ u ′ dµ and Z ∞ ( Lϕ ) dµ = − Z ∞ ϕ ′ ( x ) u ′ ( x ) dµ = Z ∞ ( ϕ ′′ ( x )) dµ + Z ∞ (cid:18) ∂ xx w + p − x (cid:19) ( ϕ ′ ( x )) dµ. Let us abbreviate w ′′ = ∂ xx w (0 , x ) , E = R udµ and also h f i = R ∞ f ( x ) dµ ( x ). Then, by usingthe above identities and by completing three squares (marked by wavy underline),Var µ ( u ) = − h u ′ ϕ ′ i − h ( Lϕ ) i = D − ϕ ′ (cid:16) u ′ − ux (cid:17)E ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ − (cid:28) ϕ ′ ux (cid:29) − (cid:28) ( ϕ ′′ ) + w ′′ ( ϕ ′ ) ✿✿✿✿✿✿✿ + p − x ( ϕ ′ ) (cid:29) ≤ (cid:28) w ′′ (cid:16) u ′ − ux (cid:17) (cid:29) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ − (cid:28) ϕ ′ ( Lϕ + E ) x (cid:29) − (cid:28) ( ϕ ′′ ) + p − x ( ϕ ′ ) (cid:29) = (cid:28) w ′′ (cid:16) u ′ − ux (cid:17) (cid:29) + (cid:10) ϕ ′′ ϕ ′ /x (cid:11) ✿✿✿✿✿✿✿✿✿ − (cid:28) ϕ ′ Ex + ( ϕ ′′ ) ✿✿✿✿✿ + ( p + 1) ( ϕ ′ ) x (cid:29) ≤ (cid:28) w ′′ (cid:16) u ′ − ux (cid:17) (cid:29) − * ϕ ′ Ex + p ( ϕ ′ ) x ✿✿✿✿✿✿ + ≤ (cid:28) w ′′ (cid:16) u ′ − ux (cid:17) (cid:29) + E p , and (28) is proven. (cid:3) References [1] S. Artstein-Avidan, B. Klartag, and V. D. Milman,
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