Dimensionality and the stability of the Brunn-Minkowski inequality

Introduction

The Brunn-Minkowski inequality states, in one of its normalizations, that

The literature contains various stability estimates for the Brunn-Minkowski inequality, which imply that when there is almost-equality in (1), then $K$ and $T$ are almost-translates of each other. Such estimates appear in Diskant , in Groemer , and in Figalli, Maggi and Pratelli . We recommend Osserman for a general survey on the stability of geometric inequalities.

The present stability estimates do not seem to imply much about the proximity of $K$ to a translate of $T$ under the assumption (2). Only if the constant “$5$” in (2) is replaced by something like $1+1/n$ or so, then the results of Figalli, Maggi and Pratelli can yield meaningful information. The goal of this note is to raise the possibility that the stability of the Brunn-Minkowski inequality actually improves as the dimension increases. In particular, we would like to deduce from (2) that

for a family of non-negative functions $p$, when the dimension $n$ is high. Here, $b_{K}$ and $b_{T}$ denote the barycenters of $K$ and $T$ respectively. Furthermore, in some non-trivial cases we may conclude (3) even when the constant “$5$” in (2) is replaced by an expression that grows with the dimension, such as $\log n$ or $n^{\alpha}$ for a small universal constant $\alpha>0$.

In this note we take the first steps towards a dimension-sensitive stability theory of the Brunn-Minkowski inequality. First, let us focus on the simplest case in which $p(x)$ in (3) is a quadratic polynomial. In fact, we are interested mainly in expressions related to the quadratic form

Let $p(x)=p_{K}(x)$ be the inertia form of $K$ defined in (4) and (5). Then,

Here $C,\alpha_{1},\alpha_{2}>0$ are universal constants, and $b_{K},b_{T}$ are the barycenters of $K,T$ respectively.

See Theorem 4.5 below for explicit bounds on the universal constants $\alpha_{1},\alpha_{2}$ from Theorem 1.1. Our interest in the inertia form $p_{K}$ stems from the central limit theorem for convex sets, see for background reading. As we shall explain in Proposition 6.4 below, Theorem 1.1 implies the bound

where $\sigma_{n}$ is the thin shell parameter from , $C>0$ is a universal constant and $\alpha_{1}>0$ is the constant from Theorem 1.1. In fact, Theorem 4.5 and (51) below show that the inequality (7) is essentially an equivalence. Consequently, the universal constant $\alpha_{1}$ from Theorem 1.1 is intimately connected with the thin shell parameter $\sigma_{n}$. The question of whether $\sigma_{n}$ is bounded by a universal constant is currently one of the central problems in high-dimensional convex geometry.

In fact, the assumption that $\varphi$ is $1$-Lipschitz may typically be weakened. For instance, when $\varphi$ is convex or concave, it is well-known that

(in order to use (8) we also need a crude estimate for $\int_{T}|x|^{2}d\mu_{T}(x)$, hence we applied Corollary 2.4 to obtain such an estimate). In view of (11) and Proposition 6.4 below, we match (up to logarithmic factors) the best bounds for the width of the thin spherical shell for unconditional convex bodies proven in .

The structure of the remainder of this note is as follows: In the next section we establish some well-known facts about one-dimensional log-concave measures. In Section 3 we prove Theorem 1.1 and in Section 4 we prove Theorem 1.2. Section 5 is dedicated to attaining some inequalities related to one-dimensional transportation of measure. In Section 6, using these inequalities, we prove Theorem 1.3.

Background about log-concave densities on the line

A nice characterization of log-concavity that we learned from Bobkov is that $\mu$ is log-concave if and only if the function

is a concave function. This characterization lies at the heart of the proof of the following Poincaré-type inequality which appears as Corollary 4.3 in Bobkov :

Let $\mu$ be a log-concave probability measure on the real line, and set

for the variance of $\mu$. Then for any smooth function $f$ with $\int fd\mu=0$,

Further information about log-concave densities on the line is provided by the following standard lemma.

$\displaystyle f(t)\leq\frac{C}{\sigma}\exp(-c|t-b|/\sigma)$; and

If $|t-b|\leq c\sigma$, then $\displaystyle f(t)\geq\frac{c}{\sigma}$.

Proof: Part (a) is the content of Lemma 3.2 in Bobkov . In order to prove (b), we show that for some $t_{0}\geq b+c_{0}\sigma$,

with $c_{0}=1/(10C)$, $C_{1}=c^{-1}\log(10C/c)$ where here $c,C$ are the constants from part (a). Indeed, if there is no such $t_{0}$, then from (a),

in contradiction to Grünbaum’s inequality (see, e.g., [4, Lemma 3.3]). By symmetry, there exists some $t_{1}\leq b-c_{0}\sigma$ with

From log-concavity, $f(t)\geq 1/(10C_{1}\sigma)$ for $t\in[t_{1},t_{0}]$, and (b) is proven since $[t_{1},t_{0}]\supseteq[b-c_{0}\sigma,b+c_{0}\sigma]$.

The following lemma is essentially a one-dimensional, functional version of Theorem 1.1. The Lemma states, roughly, that if the supremum-convolution of two log-concave probability densities has a bounded integral, then their respective variances cannot be too far from each other.

Let $X,Y$ be random variables with corresponding densities $f_{X},f_{Y}$ and variances $\sigma_{X}^{2},\sigma_{Y}^{2}$. Assume that $f_{X}$ and $f_{Y}$ are log-concave. Define

a supremum-convolution of $f_{X}$ and $f_{Y}$. Then,

Proof: The function $h$ is clearly measurable (it is even log-concave). It follows from Lemma 2.2(b) that there exist intervals $I_{X},I_{Y}$ such that

Combining this with (13), we learn that there exists an interval $I_{Z}$ with $Length(I_{Z})\geq c(\sigma_{X}+\sigma_{Y})/2$ such that,

In order to prove (14), it suffices to show that

Denote the respective densities of $X,Y$ by $f_{X},f_{Y}$. The Prékopa-Leindler theorem (see, e.g., the first pages of Pisier ) implies that $f_{X}$ and $f_{Y}$ are log-concave. Furthermore, using the Prékopa-Leindler theorem again we derive,

Plugging this into lemma 2.3 we deduce (15).

Next, for a measure $\mu$ and measurable sets $A,B$ with $0<\mu(A)<\infty$ define

Thus the probability measure $\mu|_{A}$ is the conditioning of $\mu$ to the set $A$. Clearly, for a log-concave measure $\mu$ and an interval $I$, the measure $\mu|_{I}$ remains log-concave.

Proof: It is enough to prove the lemma for $J_{1},J_{2}$ being rays. Denote by $I$ the interior of the support of $\mu$, and by $\rho$ the density of $\mu$. Abbreviate $\Phi(t)=\mu\left((-\infty,t]\right),\ \mu_{t}=\mu|_{(-\infty,t]}$ and set

To prove the lemma, it suffices to show that $v^{\prime}(t)\geq 0$ for any $t$, or equivalently, that

Deriving a stability estimate from the central limit theorem for convex sets

A second ingredient will be a calculation which shows that the integral of the supremum-convolution of two Gaussian densities whose covariance matrix is a multiple of the identity, becomes very large when their respective covariances are not close to one another. This will imply that when $Vol_{n}((K+T)/2)$ is not large, the covariance matrices of both marginals are roughly the same multiple of the identity. Therefore the inertia forms of $K$ and $T$ must have had roughly the same trace (the trace of the matrix will determine the multiple of the identity).

for all $x\in E$ with $|x|\leq n^{c_{4}}$. Here, $C,c_{1},c_{2},c_{3},c_{4}>0$ are universal constants.

It can be seen directly from the proof in that the constants in Theorem 3.1 may be selected to be $c_{1},c_{2},c_{3}=\frac{1}{30},c_{4}=\frac{1}{60},C=500$. Other constants would imply different universal constants in Theorem 1.1. We shall need the following elementary lemma:

for $\alpha=\sqrt{1/a}$ and also for $\alpha=a$, where $c>0$ is a universal constant.

Proof: First we prove the lemma for $\alpha=a$. Note that for $0<a\leq 4$,

The case where $\alpha=\sqrt{1/a}$ follows as $\min\{(\sqrt{1/a}-1)^{2},1\}\leq 10\min\{(a-1)^{2},1\}$.

The following lemma is the second ingredient in our proof of Theorem 1.1 described above. The essence of the lemma is that the integral of the supremum-convolution of two spherically-symmetric Gaussian densities must be quite large when the covariances are not close to each other.

whenever $|x|\leq 10\alpha\sqrt{k}$. Assume that $h$ is measurable. Then,

We would like to find $s$ which maximizes the right-hand side in (20). We select $s=t(a-1)/(a+1)$ and verify that when $|t|<5\sqrt{(1+a)k/a}$ we have $|s+t|\leq 10\sqrt{k}$ and $|s-t|\leq 10\alpha\sqrt{k}$. We conclude that for any $|t|<5\sqrt{(1+a)k/a}$,

for some universal constants $C,C_{1}>1$.

Proof: Clearly, we may assume that the sequence $\{\lambda_{i}\}$ is non-decreasing. Translating $g$, we may assume that the barycenter of $g$ is at the origin. Let $X$ and $Y$ be random vectors that are distributed according to the laws $f,g$, respectively. Fix $0<\delta<1$. Consider the subspace $E$ spanned by $\{e_{i};\lambda_{i}-1\geq\delta\}$, where $\{e_{i}\}$ is an orthonormal basis of eigenvectors corresponding the the eigenvalues $\{\lambda_{i}\}$. Denote $d=\dim E$ and assume that $d\geq 2$. Since the $\lambda_{i}$’s are in increasing order, the subspace $E$ has the form,

for some $1\leq i_{0}\leq n$. Write $j_{0}=\left\lfloor\frac{n-i_{0}}{2}\right\rfloor$ and $V^{2}=\lambda_{i_{0}+j_{0}}$. Now, fix $1\leq j\leq j_{0}$. Define,

Inspect the function $f(\theta)=\langle Cov(g)v_{j}(\theta),v_{j}(\theta)\rangle$. We have $f(0)=\lambda_{i_{0}+j_{0}-j}\leq V^{2}$ and $f(1)=\lambda_{i_{0}+j_{0}+j}\geq V^{2}$. By continuity, there exists a certain $0\leq\theta_{j}\leq 1$ for which

Equation (21) and the fact that $e_{1},\ldots,e_{n}$ are orthonormal eigenvectors imply that for every $v\in F$, one has $\langle Cov(g)v,v\rangle=V^{2}$. Moreover, $\dim F=j_{0}\geq\frac{1}{2}d-1$. We now apply Theorem 3.1 which claims that if $d\geq C$, then there exists a subspace $G\subset F$ with $\dim G=\lfloor d^{1/40}\rfloor$ such that

On the other hand, we may use the Prekopá-Leindler inequality as in (16) above, and deduce that

Consequently, under the assumption that $d\geq C$,

Since $V\geq\sqrt{1+\delta}\geq 1+\delta/3$, we conclude

By repeating the argument, with the subspace $\{e_{i};\lambda_{i}-1\leq-\delta\}$ replacing the subspace $E$, we conclude the proof.

Proof of Theorem 1.1: By applying affine transformations to both $K$ and $T$, we can assume that both bodies have the origin as their barycenter, and that $p_{K}(x)=|x|^{2}$ while $p_{T}(x)=\sum_{i}x_{i}^{2}/\lambda_{i}$. By Lemma 3.4,

for any $0<\delta<1$. Since $\lambda_{i}\leq CR^{4}$ for all $i$, as follows from Corollary 2.4, then

where $C,\alpha_{1},\alpha_{2}>0$ are universal constants. To obtain (6), note that

Remark: When $K$ in Theorem 1.1 is isotropic, we actually prove in (24) that

where $\|A\|_{HS}^{2}=Trace(A^{t}A)$ is the square of the Hilbert-Schmidt norm of the matrix $A$.

Obtaining stability estimates using a transportation argument

the covariance matrix. Finally, we normalize this density by defining

A theorem of Brenier asserts that a convex solution to the above equation on the domain $Supp(f)=\{x;f(x)>0\}$ exists. The regularity theory developed by Caffarelli implies that the convex function $\varphi$ is smooth. For precise definitions and properties, see . The map $F=\nabla\varphi$ pushes forward the measure whose density is $f$ to the measure whose density is $g$, and is referred to as the Brenier map between the two measures. The matrix $\nabla F(x)$ is positive-definite since it has a positive determinant and it is the Hessian matrix of a convex function.

Remark. The Knothe map, used in Section 6, is in some sense a limiting case of the Brenier map. See .

The following lemma contains the central idea of this section.

where $D=D(f,g)$ and $\Lambda$ is a random variable distributed uniformly in $$.

Proof: By a standard approximation argument we may assume that $f$ and $g$ are sufficiently smooth. Denote $D=D(f,g)$ and $L(\lambda,x)=L(f,g)(\lambda,x)$. Furthermore, define,

Using the fact that $L$ is log-concave, we obtain

A simple calculation shows that the Jacobian of $M(\lambda,x)$ is

By changing variables using $M^{-1}$ and applying (30) and (31), we calculate

Applying the change of variables $x\to D^{-1/2}(x-b(f,g))$ completes the proof.

Combining this with the above lemma yields

In view of (33), we would like to have a lower bound for $v(x,y)$ in terms of $|x|^{2}-|y|^{2}$ and in terms of $|x-y|$. The following lemma serves this purpose.

and $h(\lambda)=f(\lambda)-g(\lambda)$. Then $h(1-\lambda)=h(\lambda)$ hence $COV(g(\Lambda),h(\Lambda))=0$. Consequently,

Combining (35), (36) and (37) completes the proof.

Proof of Theorem 1.2: Write $b=b(f,g)$ and $D=D(f,g)$. Substituting the result of Lemma 4.2 into (33) yields

Let $X,Y$ be the random vectors whose densities are $f,g$ respectively. By the definition of the transportation distance,

where the transportation distance between random vectors is defined to be the distance between the corresponding distribution measures. The fact that $f$ and $g$ have barycenters at the origin implies

The Cauchy-Schwartz inequality together with (38), (39) and (40) yield,

where $\|D\|_{OP}=\sup_{0\neq x}|D(x)|/|x|$ is the operator norm of $D$. From the remark to Corollary 2.4 we conclude that

The function $\lambda\mapsto K_{\lambda}(f,g)$ is log-concave and it is bounded from below by one, according to the Prékopa-Leindler inequality. Therefore,

The rest of this section aims at a better understanding of the exponents in Theorem 1.1. The next lemma exploits the second summand in our basic estimate (41).

Proof: We use the notation of the proof of Theorem 1.2. In order to establish (42), we fix $\alpha>0$, and assume that

Consequently, in order to establish (42), it suffices to show that for some universal constant $C>0$,

In view of (41), the last inequality will be concluded if we only manage to show,

The above fact follows from an application of Lemma 3.4 with $\delta=1/2$ and from the assumption that $K_{1/2}(f,g)\leq\exp(n^{c_{1}})$. Equation (42) is established, and the proof of (43) is analogous. The proof of the lemma is thus complete.

so that $\sigma_{n}\leq\tau_{n}n^{\kappa}$. Note that the thin-shell conjecture implies that $\kappa=0$ and $\tau_{n}<C$. We apply the estimate from the previous lemma for various marginals of our $n$-dimensional measures, and obtain:

where $C,C_{1}>0$ are some universal constants.

Proof: The bound (46) follows directly from the remark to Corollary 2.4. In order to establish (47), denote by $\{e_{i}\}$ the orthonormal basis of eigenvectors corresponding to the eigenvalues $\{\lambda_{i}\}$. Define

Let $E$ be the subspace with the larger dimension among these two subspaces. Then $k=\dim E\geq i/2$. Denote by $i_{0}$ the maximal $j$ for which $e_{j}\in E$. Then $k\leq i_{0}\leq i$. According to our assumption, $\dim(E)\geq(\log(2R))^{C_{1}}/2$, and hence we may apply Lemma 4.3 in the subspace $E$. Denote by $f_{E}$ and $g_{E}$ the marginals of $f$ and $g$ to the subspace $E$. Using (42) and (43) for $f_{E}$ and $g_{E}$ we obtain

where we used the fact that $K(f,g)\leq K_{1/2}(f,g)^{2}=R^{2}$ as well as the Prékopa-Leindler inequality which implies that $K_{\lambda}(f_{E},g_{E})\leq K_{\lambda}(f,g)$ for any $\lambda\in(0,1)$.

The next theorem demonstrates that the exponent $\alpha_{1}$ in Theorem 1.1 may be made arbitrarily close to $1/2-\kappa$, thus complementing the inequality (7) which goes in the opposite direction. This provides yet another piece of evidence for the close relationship between the thin shell problem and the stability of the Brunn-Minkowski inequality in high dimensions.

where $Id$ is the identity matrix. Consequently,

Proof: We may clearly assume that $Cov(\mu_{T})$ is a diagonal matrix whose diagonal is $\lambda_{1},\ldots,\lambda_{n}$, where the sequence $\{|\lambda_{i}-1|\}$ is non-increasing. Since our measures are log-concave, then we may use Lemma 4.4 and calculate

The bound (50) follows. In order to deduce (51) from (50), argue as in (25) above. The proof is complete.

Transportation in one dimension

For $j=1,2$, the map $\Phi_{j}^{-1}$ pushes forward the uniform measure on $$to$\mu_{j}$. The monotone transportation map between$\mu_{1}$and$\mu_{2}$ is the continuous, non-decreasing function

defined for $t\in Supp(\mu_{1})$. Observe that

Furthermore, $F$ is differentiable in $Supp(\mu_{1})$ and

Additionally, it is well-known (see, e.g., Villani’s book ) that

where $F$ is the monotone transportation map between $\mu_{1}$ and $\mu_{2}$ and $C>0$ is a universal constant.

We begin the proof of Proposition 5.1 with the following crude lemma.

Let $\mu_{1}$ and $\mu_{2}$ be probability measures on the real line.

If $\mu_{1}$ and $\mu_{2}$ are even, then

If $\mu_{1},\mu_{2}$ are supported on $[A,\infty)$ and $[B,\infty)$ respectively, and have non-increasing densities, then

Proof: Denote by $\delta_{0}$ the Dirac measure at the origin. Assume that $\mu_{0}$ and $\mu_{1}$ are even. By the triangle inequality for the transportation metric,

Let $\delta_{A},\delta_{B},\delta_{e}$ be the Dirac measures supported on $A,B,e$ respectively. By the triangle inequality,

Therefore, by using $W_{2}(\mu_{1},\mu_{2})\leq W_{2}(\mu_{1},\delta_{A})+W_{2}(\delta_{A},\delta_{B})+W_{2}(\delta_{B},\mu_{2})$,

Proof of Proposition 5.1: Use (52), the definition of $F$, and the fact that $\Phi_{1}^{-1}$ pushes forward the uniform measure on $$to$\mu_{1}$, in order to obtain

Recall that when $\mu_{j}$ is a log-concave measure, the function $\rho_{j}(\Phi_{j}^{-1}(t))$ is concave on $$. Denote$I_{j}(t)=\rho_{j}(\Phi_{j}^{-1}(t))$for$j=1,2$. Then$I_{1}$and$I_{2}$are concave, non-negative functions on$$, with the property that$I_{j}(t)=I_{j}(1-t)$for any$t\in$. These two functions are therefore continuous on$(0,1)$, increasing on$[0,1/2]$, and decreasing on$[1/2,1]$. Let$\varepsilon>0$ be such that

Suppose first that $\varepsilon>1/10$. In this case, from part (i) of lemma 5.2,

So whenever $\varepsilon>1/10$, the inequality (54) holds trivially for a sufficiently large universal constant $C>0$.

From now on, we restrict attention to the case where $\varepsilon\leq 1/10$. We divide the rest of the proof into several steps.

Step 1: Let us prove that there exists a universal constant $C>0$ such that

Once we prove (57), the desired bound (56) follows from (55). We thus focus on the proof of (57). Suppose that $t_{1}\in(0,1/2]$ satisfies $I_{1}(t_{1})>4I_{2}(t_{1})$. We will show that in this case

If $I_{1}(t)>2I_{2}(t)$ for all $t\in(0,t_{1})$, then $t_{1}\leq\varepsilon^{2}$ according to (55). Thus (58) holds true in this case. Otherwise, there exists $0<t<t_{1}$ with $I_{1}(t)\leq 2I_{2}(t)$. Let $t_{0}$ be the supremum over all such $t$. Since $I_{1}$ and $I_{2}$ are continuous and non-decreasing on $(0,t_{1}]$, then

Since $I_{1}$ is concave, non-decreasing and non-negative on $[0,t_{1}]$, then necessarily $t_{0}<t_{1}/2$. We conclude that $I_{1}(t)>2I_{2}(t)$ for any $t\in[t_{1}/2,t_{1}]$. From (55) it follows that $t_{1}\leq 2\varepsilon^{2}$. Therefore (58) is proven in all cases. By symmetry, we conclude (57), and the proof of (56) is complete.

Step 2: For any $0\leq T\leq\Phi_{1}^{-1}(1-2\varepsilon^{2})$ we have

where the last inequality is the content of Step 1. Denote $\nu=\mu_{1}|_{[-T,T]}$, an even log-concave probability measure. According to Lemma 2.5, we have $Var(\nu)\leq Var(\mu_{1})\leq\sigma$. Note that the function $F(t)-t$ is odd, hence its $\nu$-average its zero. Using the Poincaré-type inequality in Lemma 2.1, we see that for any $0\leq T\leq\Phi_{1}^{-1}(1-2\varepsilon^{2})$,

Step 3: Let $T_{1}=\Phi_{1}^{-1}(1-3\varepsilon^{2})$ and let $T_{2}=\Phi_{1}^{-1}(1-2\varepsilon^{2})$. We use (59) and conclude that there exists $T_{1}\leq T\leq T_{2}$ with

Denote $\nu_{1}=\mu_{1}|_{[T,\infty)}$ and $\nu_{2}=\mu_{2}|_{[F(T),\infty)}$. These are log-concave probability densities with $Var(\nu_{1})+Var(\nu_{2})\leq\sigma^{2}$. Note that we have, owing to (59),

In order to prove the lemma it remains to show that $W_{2}(\nu_{1},\nu_{2})^{2}\leq C\sigma^{2}.$ But in view of (60), the latter is a direct consequence of part (ii) in lemma 5.2: Since $T,F(T)>0$, then the log-concave densities of $\nu_{1}$ and $\nu_{2}$ are non-increasing. This completes the proof.

We thus view the function $h$ as a refined variant of the supremum-convolution of $f$ and $g$. The following proposition is a stability estimate for the Prékopa-Leindler inequality in one dimension. It may be viewed as the transportation-metric version of the $L^{1}$-stability estimates from Ball and Böröczky .

where the function $h$ is defined via (61) and $C>0$ is a universal constant.

Proof: Multiplying the functions $f$ and $g$ by positive constants, if necessary, we may assume that $\int f=\int g=1$. Indeed, neither the left-hand side nor the right-hand side of (54) is changed under such normalization. Let $F$ be the monotone transportation map between $\mu_{f}$ and $\mu_{g}$ and as before, $S(x)=(F(x)+x)/2$ for $x\in Supp(\mu_{f})$. Applying the change of variables $y=S(x)$ we see that

According to (52), we have $F^{\prime}(x)g(F(x))=f(x)$ for any $x$ in the support of $\mu_{f}$. Since $g$ is log-concave, it does not vanish in $Supp(\mu_{g})$, and hence $F^{\prime}(x)\neq 0$ for any $x\in Supp(\mu_{f})$. Therefore,

where we used Lemma 3.2(ii) in the last passage. Since $\int f=1$, then

We may thus apply Proposition 5.1 and deduce that

Unconditional Convex Bodies

where $C>0$ is a universal constant and $\mu_{f},\mu_{g}$ are the probability measures with densities $f,g$ respectively.

The main tool in the proof of Theorem 6.1 is the Knothe map from , which we define next. Let $M,f,g$ be as in Theorem 6.1. Then the support of $\mu_{g}$ is a convex set, and $g$ does not vanish in $Supp(\mu_{g})$. The Knothe map between $\mu_{f}$ and $\mu_{g}$ is the continuous function $F=(F_{1},\ldots,F_{n}):Supp(\mu_{f})\rightarrow Supp(\mu_{g})$ for which

For any $j$, the function $F_{j}(x_{1},\ldots,x_{n})$ actually depends only on the variables $x_{1},\ldots,x_{j}$. We may thus speak of $F_{j}(x_{1},\ldots,x_{j})$.

For any $j$ and for any fixed $x_{1},\ldots,x_{j-1}$, the function $F_{j}(x_{1},\ldots,x_{j})$ is non-decreasing in $x_{j}$.

It may be proven by induction on $n$ (see ) that the Knothe map between $\mu_{f}$ and $\mu_{g}$ exists, and that in fact, the three requirements above determine the function $F$ completely. Denoting $\lambda_{j}(x)=\left.\partial F_{j}(x)\right/\partial x_{j}\geq 0$, it follows from property (b) that

for any $x\in Supp(\mu_{1})$, where $J_{F}(x)$ is the Jacobian of the map $F$. Below we will also use the fact that the map $x\mapsto x+F(x)$, defined for $x\in Supp(\mu_{f})$, is one-to-one, as follows from properties (b) and (c). Set

and let $f_{n-1},g_{n-1}$ be the densities of the probability measures $\pi_{*}(\mu_{f}),\pi_{*}(\mu_{g})$, respectively. Then $f_{n-1}$ and $g_{n-1}$ are unconditional and log-concave. Write $T_{n}=F=(F_{1},\ldots,F_{n})$ for the Knothe map between $\mu_{f}$ and $\mu_{g}$, and set

Then $T_{n-1}$ is the Knothe map between $\pi_{*}(\mu_{f})$ and $\pi_{*}(\mu_{g})$. Observe that for fixed $(x_{1},\ldots,x_{n-1})\in\pi(Supp(\mu_{f}))$, the map

is the monotone transportation map between the probability densities proportional to

for $(z_{1},\ldots,z_{n-1})=T_{n-1}(x_{1},\ldots,x_{n-1})$. For $i=n-1,n$ we set

which is a one-to-one, continuous function, defined for $x\in Supp(\mu_{f})$ when $i=n$ and for $x\in\pi\left(Supp(\mu_{f})\right)$ when $i=n-1$. According to (65) and to property (b), the Jacobian $J_{S_{i}}(x)$ of the map $S_{i}$ satisfies

Since $S_{i}$ is one-to-one, then $V(f_{i},g_{i})$ is a well-defined function on a subset of $Q^{i}$. We extend $V(f_{i},g_{i})$ to the entire $Q^{i}$ by setting it to be zero outside its original domain of definition.

Let $\varphi:Q^{n-1}\rightarrow[0,\infty)$ be a measurable function. Then,

Proof: We use (65) for the Knothe map $T_{n-1}$ to conclude that

where we used (66) and (67) in the last passage. The map $S_{n-1}$ is one-to-one in the support of $f_{n-1}$. Changing variables $z=S_{n-1}(y)$ we obtain

The following lemma will serve as the induction step in the proof of Theorem 6.1.

where $C>0$ is a universal constant (in fact, it is the same constant as in Proposition 5.3).

In order to prove the lemma, it therefore suffices to show that

Recall that $t\mapsto F_{n}(y,t)$ is the monotone transportation map between the even, log-concave probability measures supported on $[-M,M]$, whose densities are proportional to $t\mapsto f(y,t)$ and $s\mapsto g(T_{n-1}(y),s)$. The variance of an even measure supported on $[-M,M]$ cannot exceed $M^{2}$. We may therefore use Proposition 5.3, together with (53), to conclude that for any $y\in\pi(Supp(\mu_{f}))$,

In particular, the right-hand side of (70) is non-negative. We use the definition (67) and integrate with respect to $y$. This yields:

where the last passage is legal according to Lemma 6.2. The desired estimate (69) follows, and the proof is complete.

Proof of Theorem 6.1: We will prove by induction on the dimension $n$ that

where $C$ is the constant from Lemma 6.3. The case $n=1$ follows from Proposition 5.3 and from the fact that the variance of an even measure supported on $[-M,M]$ cannot exceed $M^{2}$. We assume that (71) is proven for dimension $n-1$ and proceed with the proof for dimension $n$. Apply the induction hypothesis for the unconditional, log-concave probability densities $f_{n-1},g_{n-1}$ and conclude that

and (71) is proven for dimension $n$, hence for all dimensions. Using (71) and the fact that $V(f,g)\leq H(f,g)$, the theorem follows by the definition of transportation distance.

The uniform measure on a convex body is a prime example for a log-concave measure. Consequently, we may deduce Theorem 1.3 from Theorem 6.1 by using a crude “cut with a big cube” argument. The logarithmic factor of Theorem 1.3 may be an artifact of this clumsy procedure.

Proof of Theorem 1.3: Let $0\leq\gamma\leq 1/2$ be a parameter to be specified later on. For $\alpha,\beta>0$ we denote

According to Corollary 2.4, we have $Cov(\mu_{T})\leq CR^{4}$. Using Lemma 2.2 and a union bound, we deduce that

We now select $\alpha$ and $\beta$ so that

Denote by $\mu_{K}^{1}$ the uniform probability measure on $K_{\alpha}$ and similarly for $T$. By elementary properties of the transportation metric $W_{2}$, it follows that

where $Diam(K)=\sup_{x,y\in K}|x-y|$ is the diameter of $K$. It is well-known (see ) that $Diam(K)\leq Cn\sqrt{\|Cov(\mu_{K})\|_{OP}}$ and therefore,

Note that $\mu_{K}^{1}$ and $\mu_{T}^{1}$ satisfy the requirements of Theorem 6.1 with $M=\max\{\alpha,\beta\}\cdot\log n$. Denote $f(x)=1_{K_{\alpha}}(x)/Vol_{n}(K_{\alpha}),g(x)=1_{T_{\beta}}(x)/Vol_{n}(T_{\beta})$. Then,

From Theorem 6.1 and (75) we conclude that

All that remains is to select $\gamma$. In the case where $R\leq n^{2}$, we choose

and deduce the desired bound (10) from (76). In the case where $R\geq n^{2}$, we select $\gamma=1/2$ and still deduce (10). The theorem is thus proven for all cases.

for any $s>0$ with $Vol_{n}(K_{s})/Vol_{n}(K)\in[1/8,7/8]$. Then,

Proof: Standard bounds on the distribution of polynomials on high-dimensional convex sets (see Bourgain or Nazarov, Sodin and Volberg ) reduce the desired inequality (78) to the estimate

In order to prove (79), select $a>0$ such that $Vol_{n}(K_{a})=Vol_{n}(K)/4$. From (77),

For the upper bound, let $s<t$ be such that $Vol_{n}(K_{s})=3Vol_{n}(K)/4$ and $Vol_{n}(K_{t})=7Vol_{n}(K)/8$. Then, from (77),

Hence, $\max_{x\in K_{s}}\frac{|x|^{2}}{n}\leq 1+13A$, or equivalently,

It is now clear that (79) follows from (80) and (81).

References