Hessian metrics, CD(K,N)-spaces, and optimal transportation of log-concave measures

We study the optimal transportation mapping $\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the metric-measure space $M=(\mathbb{R}^d, g, \mu)$. We prove, in particular, that $M$ admits a non-negative Bakry--{\'E}mery tensor provided both $V$ and $W$ are convex. If the target measure $\nu$ is the Lebesgue measure on a convex set $\Omega$ and $\mu$ is log-concave we prove that $M$ is a $CD(K,N)$ space. Applications of these results include some global dimension-free a priori estimates of $\| D^2 \Phi\|$. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for $M$.


Introduction
This paper is motivated by the following problem. Given two probability measures µ = e −V dx and ν = e −W dx on R d let us consider the optimal transportation mapping T = ∇Φ of µ onto ν and the associated Monge-Ampère equation (1.1) We are interested in efficient estimates of the Lipschitz constant sup R d D 2 Φ or the integral Lipschitz constant R d D 2 Φ p dµ 1 p (here · is the operator norm) for some p ≥ 1. This problem has different aspects. From the regularity theory viewpoint the "best" estimate provides the highest regularity level for Φ for any given regularity assumptions on V and W . Classical regularity results provide Φ ∈ C 2,α under assumption of the Hölder continuity of V and W . The results of this type are usually available only in finite dimensions and involve constants which are very hard to control. For an account in the regularity theory of the optimal transportation and the Monge-Ampère equation (flat case) see [15], [28] (see also [1], [26], [13], [7], [21], [6], [8]).
Our motivation partially comes from the optimal transportation theory on the infinite-dimensional spaces, in particular, on the Wiener space ( [4], [5], [12], [19]). Note that the finite-dimensional regularity techniques can not be applied here and the general regularity problem for optimal transportation on the Wiener space is open. Some partial results see in [5].
Another type of problems is studied in convex geometry. Given the target measure ν (a typical example: ν is the normalized Lebesgue measure on a convex set) find a "nice" source measure µ with known spectral properties (say, known Poincaré constant) and small L p (µ)-norm of D 2 Φ . A classical example is given by a Caffarelli's contraction theorem. According to this result every optimal transportation mapping ∇Φ pushing forward the standard Gaussian measure onto a log-concave measure ν with the uniformly convex potential W (i.e. D 2 W ≥ K · Id with K > 0) is a 1 Let us recall a related open problem. A convex set Ω is called isotropic if IE(x i ) = 0, IE(x i x j ) = δ ij (here IE means the expectation with respect to the normalized Lebesgue volume on Ω). The Poincaré constant is the minimal constant c p such that for any smooth f . According to the famous Kannan, Lovász, and Simonovits conjecture (KLS conjecture) the Poincaré constant of any isotropic convex set is bounded by some universal number. This is one of the most difficult open problems naturally arising in convex geometry (KLS conjecture, slicing problem, thin-shell conjecture).
By a classical result of Payne and Weinberger (see [25]) c p ≤ diam 2 (Ω) π 2 for any convex Ω. Thus in view of the Caffarelli's theorem it is natural to expect that the Lipschitz constant of the optimal transportation mapping pushing forward, say, the standard Gaussian measure γ onto a convex set Ω is controlled at least by the diameter of Ω. However, even this turns out to be difficult to prove. According to [17] this Lipschitz constant is controlled by √ d diam(Ω). It is still not known whether D 2 Φ dγ ≤ C diam(Ω) for some universal C. We prove in this paper that where Λ(x) = D 2 Φ(x) , µ = γ is the standard Gaussian measure, c is universal, and ν is the normalized Lebesgue measure on Ω.
We apply here a geometric approach developed by E. Calabi in [9] in his study of the regularity problem for the Monge-Ampère equation. The sharpest regularity results for the Monge-Ampère equation have been obtained later by other methods, including the Krylov-Safonov-Evans regularity theory. Let us cite Nikolai Krylov (see [21]). "To prove the existence of solutions of equations like (1.1) by the methods known before 1981 was no easy task. It involved finding a priori estimates for solutions and their derivatives up to third order. Big part of the work is based on differentiation (1.1) three times and on certain extremely cleverly organized manipulations invented by Calabi. After 1981 the approach to fully nonlinear equations changed dramatically." Unfortunately, the deep techniques developed by Krylov, Safonov, Evans, Caffarelli and others don't work in applications where some dimension-free bounds needed. These applications include convex geometry and analysis on Wiener space. This is the main reason why we come back to the old Calabi's trick.
Let us explain the main idea of this approach. Differentiating (1.1) along a vector e one obtains the following quasilinear diffusion equation where L Φ f = div ν ∇f (∇Ψ) • (∇Φ) and Ψ = Φ * is the Legendre transform of Φ. This diffusion operator is a generator of the following symmetric Dirichlet form: It is natural (and it was the Calabi's observation) to introduce the following Riemannian metric on R d : g ij = Φ x i x j (in the fixed initial coordinate system). The corresponding manifold will be denoted by M . The manifolds of this type are called "Hessian manifolds" and they are real analogs of the complex Kähler manifolds. More on Hessian manifolds see in [10], [27]. The Dirichlet form E Φ can be rewritten as follows: where ∇ M is the gradient on M . Note that µ is not the Riemannian volume vol M measure of M . In fact where P = 1 2 V + W (∇Φ) . We introduce the metric-measure space (R d , g, µ) denoted by the same letter M .
The diffusion generator can be rewritten in geometric terms We note that the inverse optimal mapping ∇Ψ = (∇Φ) −1 defines a dual metric-measure space M ′ = (R d , D 2 Ψ, ν) and the mapping x → ∇Φ(x) is a measure preserving isometry between M and M ′ .
In this paper we compute the second "carré du champ" operator Γ 2 which is responsible for spectral properties of the Dirichlet form E Φ . Equivalently, we compute the Bakry-Émery tensor Ric + D 2 M P of the metric-measure space (R d , g, µ). In particular, we get Theorem 1.1. Assume that V and W are convex. Then the Bakry-Émery tensor of M = (R d , g, µ) is non-negative.
Further we investigate the concentration properties of M . To this end we establish the following elementary but useful inequality where d is the standard Euclidean distance. We are especially interested in a particular case when the source measure µ is log-concave and the target measure ν is the Lebesgue measure on a convex set A. It turns out that in this case one can establish a stronger result than Theorem 1.1. We prove namely that M belongs to the family of the so-called CD(K, N ) spaces. Theorem 1.2. Assume that µ = e −V dx is a log-concave measure and ν is the Lebesgue measure on a convex set A. Then M is a CD(0, 2d)-space.
If, in addition, The metric-measure spaces satisfying CD(K, N )-condition are widely studied in analysis. The most significant applications of this concept are deeply related to the optimal transportation theory (see [29]). Roughly speaking, these spaces have analytical and geometrical properties comparable to those of the corresponding model spaces (spheres and hyperbolic spaces). The list of properties which can be extracted from the CD(K, N ) property is rather impressive. These are Sobolev and isoperimetric inequalities, diameter bounds, volume growth of balls, Laplacian comparison estimates etc. In particular, the general theory implies several interesting results in our special situation. Applying inequality (1.3) to the case when ν is the normalized Lebesque measure on a convex set Ω and µ = γ is the standard Gaussian measure, we get that M has a superquadratic (fourth order) concentration function. It is known that the Gaussian-type concentration together with an appropriate CD(K, N )-condition implies that M has a bounded diameter. More precisely, we get Theorem 1.3. Let µ = γ be the standard Gaussian measure and ν be the Lebesgue measure on a convex set with diameter D. Assume, in addition, that M is geodesically convex. There exists a universal constant C > 0 such that We recall that a smooth (non-complete) Riemannian manifold M is called geodesically convex if with every two points x, y ∈ M it contains a smooth geodesic path γ : [0, T ] → M joining x and y: γ(0) = x, γ(T ) = y in such a way that γ is the shortest way from x to y. By the Hopf-Rinow theorem every complete manifold is geodesically convex. It seems that the assumption of geodesical convexity of M in Theorem 1.3 can be omitted, but we were not able to prove this.
According to a recent result of E. Milman [23] the concentration and isoperimetric inequalities are equivalent in the case of the positive Bakry-Émery tensor. In particular, applying our concentration estimates, we get that the Poincaré constant of M depends on the diameter of Ω only. Then applying techniques developed in [18] we prove that if µ is standard Gaussian and ν normalized Lebesgue on a convex set, then Then (1.2) follows from the Poincaré inequality for (R d , g, µ). We also establish certain reverse Hölder inequalities for Λ. Applications of the Hessian structures in convex geometry can be found in a recent paper of Bo'az Klartag and Rohen Eldan [11]. In particular, it was shown in [11] that the positive solution to the thin shell conjecture implies the positive solution to the slicing problem. Applications of the Kähler metrics to the Poincaré-type inequalities and thin-shell estimates can be found in [16]. It was pointed out to the author by Klartag that some of the results from [16] can be generalized by the methods obtained in this paper. More on Kähler manifolds and convex sets see in [14].
The author is grateful to Emanuel Milman, Bo'az Klartag, and Ronen Eldan for stimulating discussions during his visit to the Technion university of Haifa and the Tel-Aviv University.
This research was carried out within "The National Research University Higher School of Economics" Academic Fund Program in 2012-2013, research grant No. 11-01-0175 and supported by the RFBR projects 10-01-00518, 11-01-90421-Ukr-f-a, and the program SFB 701 at the University of Bielefeld.

Diffusion viewpoint
We consider the optimal transportation mapping ∇Φ pushing forward a probability measure µ = e −V dx onto another probability measure ν = e −W dx. In order to avoid unessential technicalities we assume in this section that V , W , Φ, and are smooth functions on the whole R d .
In particular, ∇Φ, ∇Ψ are reciprocal mappings: ∇Ψ • ∇Φ(x) = x and the Hessians of Φ, Ψ satisfy the following identity D 2 Φ(x) · D 2 Ψ(∇Φ(x)) = Id for every x. This means, in particular, that D 2 Φ, D 2 Ψ are always non-degenerate (positive) matrices. By the change of variables formula Let us differentiate this formula along the vector e ∈ R d . Everywhere below the partial derivative of f along e ∂f ∂e (x) = lim t→0 f (x + te) − f (x) t will be denoted either by ∂f ∂e or (for brevity) by f e . Thus, we use breve notations ∇f e , D 2 f e for ∇ ∂f ∂e , D 2 ∂f ∂e .
After these agreements we can write the result of differentiating of the change of variables formula in the following form: Let us consider the following diffusion operator: Lemma 2.1. The following identity holds for any f, h ∈ C ∞ 0 (R d ): Proof. Let us apply the relations ∇Ψ = (∇Φ) −1 and Note that Conclusion: Φ e satisfies the following quasilinear diffusion equation: where L Φ is the generator of the Dirichlet form It is well-known that the second "carré du champ" operator Γ 2 introduced by D. Bakry (see, for instance, [2]) is responsible for spectral properties of the corresponding diffusion where Γ Φ f = (D 2 Φ) −1 ∇f, ∇f . We will compute Γ 2 in Sections 3-4.
Example 2.2. Computation of Γ 2 in the one-dimensional case: Consider a Dirichlet form One has We get from this equation that Substituting this into the formula for Γ 2 , one can easily obtain

Differential-geometric viewpoint.
Recall that the Hessian of a smooth function f on a Riemannian manifold M is the tensor defined on a couple of vector fields X, Y by the following formula where Γ j ik are the corresponding Christoffel symbols. Following E. Calabi [9] we consider a Riemannian metric g on R d given by the Hessian of the function Φ (with respect to the standard Euclidean connection). In a fixed standard orthogonal coordinate system one has In the computations below we follow the standard geometric agreements: (Einstein summation) the expressions A i B i , A ij B ij etc. mean that one takes a sum over repeating indexes: The Riemannian gradient can be computed as follows: Our manifold M belongs to the class of the so-called Hessian manifolds (see [10], [27]), which are real analogs of the Kähler manifolds intensively studied in differential geometry. Some of the computations below can be found in [9] or [27] but we give them for completeness of the picture. All the objects related to M (considered as a Riemannian manifold) will be written with the subscript M : ∇ M is the gradient, D 2 M is the Hessian, and ∆ M is the Laplace-Beltrami operator on M .
Let us rewrite the Dirichlet form in geometric terms: Computation of the Riemannian volume In what follows we compute the Ricci tensor of g. The Christoffel symbol For computing the Ricci curvature tensor we apply the following well-known formula for the Riemannian tensor with lowered indexes: The first part of this expression vanishes and we get Now we take into account the Monge-Ampère equation. Differentiating we get another version of (2.6): Finally we get the following expression for the Ricci tensor: Note that the first part defines a non-negative quadratic form:

Γ 2 -operator and geometric properties of M
In this section we calculate the second carré du champ operator Γ 2 Applying formula (3.8) which represents the Dirichlet form E Φ via the energy integral over M equipped with the measure µ = e −P dvol we rewrite the diffusion operator L Φ as follows: We apply here the Bochner's identity A generalization of this formula to the metric-measure spaces (see, for instance, [29]) gives the following expression for Γ 2 : Recall that the quantity R ∞,µ := Ric + D 2 M P is called the Bakry-Émery tensor. The notation R ∞,µ will be explained in the very last section of the paper.
The Bakry-Émery tensor has been introduced in [3]. According to a classical result of Bakry andÉmery the positivity of this tensor implies the log-Sobolev inequality for manifolds with measures.
Let us compute Ric + D 2 M P . Applying the coordinate expression for the Hessian (3.7) we get Differentiating the relation D 2 Φ · D 2 Ψ(∇Φ) = Id one can easily obtain than for every vector e the following identity holds With the help of all these computations one can easily verify that the Hessian of a smooth function f has the following symmetric expression: Writing down the expressions for the Hessians of V and W (∇Φ) and applying (3.9) one can easily get Corollary 4.2. The Bakry-Émery tensor R ∞,µ has the following coordinate expression Finally we obtain Theorem 1.1 and even more In particular, if V and W are convex, when R ∞,µ ≥ 0.

Concentration and isoperimetric properties of M
We denote by d the standard Euclidean distance and by d M the distance on the manifold M . Almost everywhere below we deal with log-concave measures. Recall that the probability measure with Lebesgue density µ = e −V dx is called log-concave if V is a convex function. Note that this function may take value +∞ outside of a convex set A. A function β i ∈ L 1 (µ) is called logarithmic derivative of µ along x i is the following identity holds If V is regular enough, the only reasonable candidate for β i is V x i . If V x i is well-defined and integrable with respect to µ, then this is indeed the case. However, not every log-concave measure has a logarithmic derivative. It is easy to check that the normalized Lebesgue on a compact convex set does not have any logarithmic derivative.
In this section we study concentration properties of the manifold M under assumption that the target measure ν is compactly supported. They easily follows from the concentration properties of µ with the help of the following lemma. A similar result see in [11] (Lemma 3.2).
Lemma 5.1. The following estimate holds Proof. Take two points x, y and join them with the line t → x + tv, where v = y−x d(x,y) . By the definition of the Riemannian distance one has Corollary 5.2. Assume that ν has a bounded support diam(supp(ν)) = D. Then for every A ⊂ R d one has According to the previous lemma In particular, one can estimate the concentration function of the metric-measure space (M, d M , µ). Recall that a function K µ is called concentration function for µ if it satisfies the following inequality: 2 and any h ≥ 0. In particular, if diam(supp(ν)) = D and µ is standard Gaussian, then In the absence of the uniform bound some concentration estimates are still available. Proof.
The very last inequality follows from the definition of the concentration function This means, in particular, that the concentration function of our metric-measure space M can be estimated by (5.11) Till the end of the section we deal with the log-concave target and source measures. It is well known (the most general statement of this type has been obtained by Emanuel Milman [23]) that concentration inequalities imply isoperimetric inequalities under assumption of the positivity of the the Bakry-Émery tensor. More precisely Theorem 5.4. [E. Milman] Let M be a smooth complete oriented connected Riemannian manifold equipped with the measure µ = e −P dvol. Assume that the corresponding Bakry-Emery tensor is nonnegative and the concentration function K M satisfies K M (r) ≥ α(r), ∀r ≥ α −1 (log 2). Then the isoperimetric function I M (t) of M can be estimated from below by where the constant c 1 is universal and c 2 depend solely on α.
We apply this theorem and the concentration inequalities obtained above to get isoperimetric inequalities on M . We have to overcome on this way certain technical difficulty: eventual noncompleteness of M . In the following lemma we establish sufficient conditions for a Hessian manifold to be complete, but these conditions are not always fulfilled in the applications we consider.
Lemma 5.5. Assume that V and W are smooth functions, defined on the whole R d , C = sup x D 2 W (x) < ∞ and there exists c > 0 such that D 2 V (x) ≥ c · Id for every x ∈ R d . Then M is unbounded and complete.
Proof. It is sufficient to show that there exists ε > 0 such that d M (x, y) ≥ εd(x, y). It follows from the Caffarelli-type estimate (see, for instance, [20]) that the norm of D 2 Ψ , where Ψ = Φ * , is uniformly bounded by C c . Since ∇Φ and ∇Ψ are reciprocal and The assumptions of Lemma 5.5 are very restrictive. However, it is possible to prove some isoperimetric-type estimates in the situation when M is not complete.
Proposition 5.7. Let µ = γ be the standard Gaussian measure and ν be a log-concave measure with bounded support Ω, where diam(Ω) = D. There exists an universal constant c such that M satisfies the following Poincaré inequality where diam(Ω) = D for every locally Lipschitz function f : R d → R with a bounded (in the standard Euclidean metric) support.
Step 1. Let us construct a sequence of smooth convex functions W n with the following properties: (1) the measures ν n = e −Wn dx converge weakly to ν, (2) every W n has uniformly bounded second derivatives, (3) for every n > 0 there exists a number N (n) such that D 2 W k ≥ n · Id on {x : |x| ≥ 2D} for k > N (n). This type of construction is quite standard and we omit here the details. We need to show that the corresponding sequence (or just a subsequence) of optimal transportations ∇Φ n converges in a sense to ∇Φ. In fact, we show the following a priori estimate: sup n D 2 Φ n 2 dγ < ∞. Then it follows immediately from the compactness embedding theorem (applied to local Sobolev spaces) and convexity of Φ n that there exists a subsequence (denoted again by the same index) such that 1) ∇Φ n → ∇Φ almost everywhere, 2) ∂ Let us show that the desired estimate holds indeed. We apply the following a priori bound proved in [18] (see section 9 below) which holds for every sufficiently regular measures e −V dx, e −W dx and the corresponding optimal transportation ∇Φ. The direct application of this inequality gives, however, nothing, because the sequence D 2 W n is not supposed to be uniformly bounded from below by a positive matrix. Let us do the following trick: we apply this inequality to the measuresμ n = e −P (|∇Φn| 2 /2) ·γ andν n = e −P (x 2 /2) · ν n with some convex function P to be chosen later. Note that ∇Φ n pushes forwardμ n ontoν n , hence estimate (5.13) is applicable. One obtains By the Cauchy-Bunyakovsky inequality for every couples of vectors a, b there exists C ε such that Thus Applying the identity We prove the desired estimate if we find a bounded P such that for some c > 0. To this end we choose for P any smooth non-decreasing function such that P (t) = δt for t ≤ 2D 2 and P (t) = 2D 2 δ + 1 for t ≥ 3D 2 . Choosing a sufficiently small δ, sufficiently big n, and taking into account property 3) of W n we easily conclude that (5.14) holds at least starting from some number n 0 . Indeed, for carefully chosen parameters the part depending on P is uniformly bounded from below for t < 2D 2 and exceeds − sup n≥n 0 ,|x|≥2D D 2 W n (x) for t ≥ 2D 2 . Finally, using that 1 ≤ P ≤ 2D 2 δ + 1, we get where · HS is the Hilbert-Schmidt norm. We note that Φ is a smooth function as far as W n is smooth inside of the support Ω of the measure ν. This follows from the regularity theory for the Monge-Amperè equation (see [18] for details).
Step 2. The property 2) of approximating measures ensures that every metric-measure space M n = (R d , D 2 Φ n , γ) is complete (Lemma 5.5) and has a non-negative Bakry-Émery tensor. Theorem 5.4 is applicable and we conclude that every M n satisfies an isoperimetric inequality defined by its concentration function K Mn . In particular, one has for every locally Lipschitz function f where C n is (any) constant satisfying I Mn (t) ≥ C n t, 0 ≤ t ≤ 1 2 , I Mn is the isoperimetric function of M n , and c is a universal constant. It remains to prove that we get (5.12) in the limit.
We apply (5.11) to estimate K Mn . Choosing t to be equal to D in (5.11) one can easily see that the term K νn (D) tends to zero. This follows from the weak convergence and the fact that ν is supported on the set Ω ⊂ {x : |x| ≤ D}. This means that the concentration function of the limiting space is estimated from below by K γ (h 2 /D). Clearly, K M (h) ≥ c h 4 D 2 for some universal c. Let us apply a simple rescaling argument. Consider the new metricd M = 1 The metric-measure spaceM = (R d ,d M , γ) still has a non-negative Bakry-Émery tensor and its concentration function is bigger than ch 4 . Hence its Poincaré constant can be estimated by some universal number. Then it follows immediately that C = lim n C n ∼ 1 √ D . Since Φ is smooth, it is sufficient to prove (5.12) for a function of the type f = g(∇Φ), where g is locally Lipschitz such that the support of g lies positive distance of ∂Ω. Let us apply (5.15) to f n = g(∇Φ n ). It remains to show that One has ∇Φ n → ∇Φ almost everywhere. The desired convergence follows immediately from the fact that ∂ The proof is complete.

Applications: bounds for integral operator norms and reverse Hölder inequalities
6.1. A variance estimate for the operator norm D 2 Φ . In this section we will apply some results and techniques developed in [20]. Denote by the operator norm of D 2 Φ. The following inequality for the operator norm has been proved in [20]. Assume that V and W are twice continuously differentiable functions. Then where L Φ Λ is understood in the distributional sense. To be more precise, for every nonnegative smooth compactly supported function η one has If there exists a global smooth field of the eigenvectors v corresponding to the largest eigenvalue Λ, this identity follows immediately from Lemma 7.1 of [20] and the relations [20]). For the full justification of this formula in the general case see the proof of Theorem 7.3 in [20]. Approximating function f (Λ) by smooth compactly supported functions, one obtains the following theorem.
Theorem 6.1. For every non-negative differentiable function f the following inequality holds In what follows we obtain some bounds on the operator norms of D 2 Φ in the case when the target measure is a log-concave measure on a bounded convex set Ω and the source measure µ = γ is Gaussian. It is an open problem whether D 2 Φ dγ ≤ Cdiam(Ω) for some universal C. We obtain below some related results. Theorem 6.2. Let µ = γ be the standard Gaussian measures and ν be a log-concave measure on a bounded convex set Ω. There exists an universal constant c such that where Λ is the operator norm of D 2 Φ, and diam(Ω) = D.
Proof. The formal proof can be obtained by applying Theorem 6.1 to f = 1. One obtains inequality Then the result follows from the Poincaré inequality for M proved in (5.12). Note, however, that (5.12) is proved for compactly supported functions only, which is not the case with Λ. To avoid this difficulty we approximate ν (in the same way as in the Proposition 5.7) by smooth log-concave measures ν n = e −Wn dx with uniformly bounded second derivatives. We get (6.17) for M n (note that it is applicable to locally Lipschitz functions, see the proof Proposition 5.7).
By the Poincaré inequality we get boundedness of c n Λ n dµ − √ Λ n dµ 2 , where c n are Poincaré constants of (R d , D 2 Φ n , γ). In the same way as in Proposition 5.7 we get in the limit 6.2. Reverse Hölder inequalities. We conclude this section by a brief discussion of the reverse Hölder inequalities for Λ. Application of Theorem 6.1 to the powers of Λ gives a kind of reverse Hölder inequalities for p ≥ 0: It is also possible to obtain certain reverse Hölder inequalities with universal constants. To this end let us note that We come back to the standard Gaussian measure and Euclidean structure. One gets Apply the Poincaré inequality for the standard Gaussian measure: One gets that for p satisfying 4(p + 1) > p 2 there exists C p such that

Computations of higher order and Calabi-type estimates
In this section we give a list of formulas which can be useful for further investigations. In particular, in this section we generalize the classical Calabi's computations from [9] (see also [8]). We don't give any specific applications of this, but we believe that this may be an important technical tool for many other problems. For instance, the Calabi estimates were in the heart of the Yau's approach to the complex Monge-Ampère equation in his solution of the Calabi's problem. 7.1. Basic quantities. Unlike the Calabi's approach we deal with the diffusion operator L Φ and don't use neither covariant derivatives nor geometrical identities. Nevertheless, we use the language of Riemannian geometry which is very well adapted for this computations.
The functions V , W and Φ are supposed to be smooth. We use the standard summation convention. All the computation are made in the fixed (global) chart and g = D 2 Φ. f m denotes the partial derivative of f with respect to x m and we set: . . . First we note that the relation V x i = −L Φ Φ x i can be rewritten as follows: It is convenient to apply the following relations in the computation Proof. One has From the relation g ij g jk = δ ik one gets This is exactly (7.19).
We get from (7.22) that g mr (g is ) mr = −g mr g ij g ks Φ jkmr + 2g mr g ij g kb g as Φ bar Φ jkm In the other hand, (7.19) implies Hence g mr (g is ) mr = g ij g ks V x j x k + g ij g ks g ar g bp Φ abj Φ rpk − g ij g ks g rj g kp W xrxp (∇Φ) − g ij g ks W xr (∇Φ)Φ jkr . Then (7.21) implies (7.20).
The following lemma is obtained by direct computations and we omit the proof here. One gets (7.23) by differentiating (7.19). The relation (7.24) follows from (7.23), (7.22), and (7.20) by the Leibnitz rule.
7.2. Calabi-type estimates. We are interested in the quantity The computation of this quantity was the main technical point of the Calabi's approach. Function R = 1 4 Φ abc Φ abc coincides with the scalar curvature for the case of constant V and W (this corresponds to the optimal transportation of the Lebesgue measure on convex sets). Calabi has shown that in this case ∆ M R ≥ C(d)R 2 . Thus R is superharmonic and this fact can be used to control the growth of R. In fact, L Φ Φ abc Φ abc controls the fourth-order derivatives too.
We compute this quantity by applying the Leibnitz rule and the Lemmata from the previous section. Let us omit the lengthy computations and state the final result.
A remarkable observation which goes back to Calabi is the following Proposition 7.3. There exists a universal constant C > 0 such that Proof. First we note that the value of the expression Φ abc Φ abc is invariant with respect to any orthogonal coordinate change. Indeed, The direct computations give immediately thatB = O * BO, where x → Ox is an orthogonal transformation andB is the corresponding matrix in the new basis. This implies the invariance. Fix a point x and choose an orthogonal basis in such a way that D 2 Φ(x) is diagonal. Denote by µ i , 1 ≤ i ≤ n the eigenvalues of (D 2 Φ) −1 . Then Rearranging the indices one can rewrite III in the following way: for every fixed a, b, c, d. It is easy to check that the function is non-negative. Hence The proof is complete.
where B is defined as above. Now it is an easy exercise to recover the following Calabi's estimate: Hence, if µ and ν are both normalized Lebesgue measure on convex sets, then for some universal c > 0 .

A remark on Kähler manifolds and convex sets
Let us assume that µ is a log-concave measure and ν is the normalized Lebesgue measure on a convex set Ω. It was shown in [16] that for any Lipschitz function f : Ω → R with Ω f dx = 0 under certain additional technical assumption on Φ (regularity at infinity) one has the following Poncaré-type inequality This inequality implies some thin-shell estimates on the simplex. An important point in the proof from [16] was the embedding of the initial space into a toric Kähler manifold. This Kähler manifold admits a nonnegative Ricci tensor. Finally, the results follow from the Bochner's identity. It was pointed out to the author by Bo'az Klartag that inequality (8.25) can be generalized to the case when ν is any log-concave measure if instead of Kähler structure one applies the metric-measure space studied in this paper and Theorem 1.1.
For applications of Hessian structures in statistics see [27] and the related references.

Hessian manifolds as CD(K, N )-spaces and diameter bounds
We recall that a smooth d-dimensional manifold equipped with a measure µ = e −P dvol, P ∈ C 2 (M ) is called CD(K, N )-space if the modified Bakry-Émery tensor In the case N = ∞ the tensor R ∞,µ coincides with the Bakry-Émery tensor. Let us show that our metric-measure space is a CD(K, N )-space.
Proof. The result follows from Corollary 4.2. It is helpful to apply the following relation Φ j mi Φ m jk = g jl g ms Φ xmx i x l Φ xsx j x k = Tr (D 2 Φ) −1 D 2 Φ x i (D 2 Φ) −1 D 2 Φ x k . By the Cauchy inequality and (7.18) Theorem 9.2. Assume that µ = e −V dx is a log-concave measure and ν is the normalized Lebesgue measure on a convex set Ω. Then M = (R d , g, µ) is CD(0, 2d)-space.
If, in addition, Proof. Recall that µ = e −P dvol M , where P = 1 2 (V + W (∇Φ)). Since W is constant on supp(ν), one gets from the previous lemma that Remark 9.3. It was pointed out to the author by Bo'az Klartag that this result can be obtained by embedding M onto the corresponding toric 2d-dimensional Kähler manifold and applying computations in C d (which are easier than in the real case). This gives a geometric interpretation of the constant 2d appearing in the CD(K, N )-condition.
It is known that the CD(K, N )-spaces satisfy the Myer's and the Bishop-Gromov comparison theorem (see [29]). But this holds in general for complete manifolds only, which is not always the case with M . Indeed, one can easily verify that if µ is the standard Gaussian measure and ν is the Lebesgue measure on a simple set (ball or cube), then M is bounded and not complete. Nevertheless, some classical results can be easily verified for the case of geodesically convex manifolds, which are manifold admitting the following property: every two points x, y ∈ M can be joined by a shortest geodesic curve γ(t) : t → exp x (tv), t ∈ [0, dist(x, y)], v ∈ T M x , |v| = 1.
In particular, the standard proof of the Bishop-Gromov comparison theorem for volume growth of balls can be easily generalized to geodesic convex spaces. Applying this fact we get immediately the following result.
Corollary 9.4. Assume that µ = e −V dx is a log-concave measure and ν is the Lebesgue measure on a convex set Ω. Assume, in addition, that M is geodesically convex. Then r → µ({x : d M (x, x 0 ) ≤ r}) r 2d is a non-increasing function.
It is known (see [29] for precise statements and for the references) that concentration inequalities together with non-negativity of the Ricci tensor imply boundedness of the manifold. Here we apply concentration arguments to our space M . Theorem 9.5. Let µ = γ be the standard Gaussian measure and ν is the Lebesgue measure on a convex set with diameter D. Assume, in addition, that M is geodesically convex. There exists a universal constant C > 0 such that Proof. We follow the arguments from [22] Theorem 7.4. Apply the concentration inequality (5.10).
Remark 9.6. There are some reasons to believe that the assumption of geodesic convexity of M can be omitted, but we were not able to prove this.