Entropy and functional forms of the dimensional Brunn–Minkowski inequality in Gauss space

Let $\gamma$ denote the standard Gaussian probability measure on $\mathbb{R}^{n}$, $\textnormal{ d}\gamma(x)\propto e^{-\frac{1}{2}|x|^{2}}\textnormal{ d}x,$ where $|\cdot|$ denotes the Euclidean norm. It was shown by Eskenazis and Moschidis [15] that, if $K_{0}$ and $K_{1}$ are origin-symmetric convex bodies, then

Here $(1-t)K_{0}+tK_{1}=\{(1-t)x_{0}+tx_{1}:x_{0}\in K_{0},x_{1}\in K_{1}\}$ denotes the collection of $t$-midpoints of all segments from $K_{0}$ to $K_{1}$. Observe that, the inequality (3) cannot hold for all compact sets $K_{0},K_{1}$. This can be easily seen by fixing a set $K_{0}$ of positive Gaussian measure, $K_{1}=\{x\}$, and sending $x\to\infty$. Without extra conditions on $K_{0}$ and $K_{1}$, the Gaussian measure only satisfies

by the virtue of being log-concave. Recall that a measure $\nu$ is said to be log-concave if it has a density of the form $\frac{\textnormal{ d}\nu}{\textnormal{ d}x}=e^{-V}$, $V$ convex, with respect to the Lebesgue measure.

The inequality (3) was conjectured by Gardner and Zvavitch [17], originally for convex bodies containing the origin. But soon after, Nayar and Tkocz [27] found a counter-example and suggested the assumption of symmetry about the origin. It must be mentioned that the work of Eskenazis and Moschidis closed the proof of (3) by verifying a sufficiency condition introduced by Kolesnikov and Livshyts [22], which is itself based on a machinery developed by Kolesnikov and E. Milman [23, 24].

More generally, the following conjecture has garnered a lot of attention in the last few years.

One reason why Conjecture 1.1 is of substantial interest is that it follows from the celebrated log-Brunn–Minkowski conjecture of Böröczky, Lutwak, Yang, and Zhang [6]. This implication was shown by Livshyts, Marsiglietti, Nayar, and Zvavitch [26]. Exciting recent developments include works by Livshyts [25], Cordero-Erausquin and Rotem [12].

Recently, Aishwarya and Rotem [1] took a completely different route to prove dimensional inequalities such as in (5) using entropy.

The technique in [1] is based on the variational principle [1, Lemma 2.7]:

which holds for every compact set $K$ and is attained by the normalised restriction $\nu_{K}$ of $\nu$ to $K$, that is, $\nu_{K}(E)=\frac{\nu(E\cap K)}{\nu(K)}$ for every Borel set $E$. As in formula (7), we will consistently write $\mathcal{P}(K)$ for the collection of all probability measures on a given $K$.

Suppose $\nu$ is a probability measure, and $Y$ is a random vector with distribution $\nu$. In light of the above variational formula, to prove the inequality 5, it suffices to show the existence of a joint distribution $(X_{0},X_{1})$ with the marginals $X_{0},X_{1}$ having distributions $\nu_{K_{0}},\nu_{K_{1}}$, respectively, such that the following entropy inequality holds:

This is because the distribution of $(1-t)X_{0}+tX_{1}$ lies in $\mathcal{P}\left((1-t)K_{0}+tK_{1}\right)$.

The definition and remarks below clarify our use of some standard terminology regarding joint distributions of random vectors.

The coupling $(X_{0},X_{1})$ used in [1] to obtain several results is the so-called optimal coupling for the Monge–Kantorovich problem with quadratic cost, namely the one that minimises $\mathbb{E}|X_{0}-X_{1}|^{2}$. For example, [1, Theorem 1.3] implies that, when $Y=Z$ has standard Gaussian distribution, the inequality (8) holds for the optimal coupling with a worse exponent ($\frac{1}{2n}$ instead of $\frac{1}{n}$) but for a larger class ($K_{0},K_{1}$ are only assumed to be star-sharped with respect to origin, not necessarily symmetric or convex). This was the first time that a dimensional Brunn–Minkowski inequality was obtained for the Gaussian measure without convexity assumptions on the admissible sets (which is not possible with the earlier approach). However, while trying to obtain an inequality of the form (8) that would strengthen the result of Eskenazis and Moschidis, the authors in [1] faced a very interesting problem.

Recall that, a random vector $X$ is said to satisfy a Poincaré inequality with constant $1$ over a class of functions $\mathcal{F}$, if for every function $f\in\mathcal{F}$, we have $\textnormal{Var}(f(X))\leq\mathbb{E}|\nabla f(X)|^{2}$. Further, a strongly log-concave random vector is one with distribution $\mu$ such that $\frac{\textnormal{ d}\mu}{\textnormal{ d}\gamma}$ is a log-concave function (in this case, $\mu$ is said to be a strongly log-concave measure). The relevance of this property in our context stems from the fact that $\gamma_{K}$ is strongly log-concave whenever $K$ is a convex body. [1, Theorem 4.5] shows that the desired inequality (8) for $Y=Z$ holds for the optimal coupling, and $X_{0},X_{1}$ even strongly log-concave, if the answer to Question 1.4 is positive.

The first main result of the present work is that there exists a coupling of even strongly log-concave random vectors such that (8) holds when $Y=Z$.

The coupling we use is not the optimal coupling, but nonetheless arises from optimal transport. Let $U,V$ be $\mathbb{R}^{n}$-valued random vectors satisfying $\mathbb{E}|U|^{2},\mathbb{E}|V|^{2}<\infty$, such that their distributions have density with respect to the Lebesgue measure. Then, a theorem of Brenier (see [29, Theorem 2.12 (ii)]) guarantees that a unique coupling minimises $\mathbb{E}|U-V|^{2}$, and furthermore, it is given by $(U,T(U))$ where $T=\nabla\phi$ is the gradient of a convex function $\phi$. Note that the map $T$, called the Brenier map from $U$ to $V$, pushes forward the distribution of $U$ to the distribution of $V$. In the present work, we consider the Brenier map $T_{0}$ from $Z$ to $X_{0}$, the Brenier map $T_{1}$ from $Z$ to $X_{1}$, and work with the joint distribution $(X_{0},X_{1})=(T_{0}(Z),T_{1}(Z))$.

The contraction theorem of Caffarelli [10] tells us that the Brenier map from the standard Gaussian to any strongly log-concave random vector is $1$-Lipschitz. This automatically gives us that the $X_{t}=(1-t)X_{0}+tX_{1}$ we consider in this paper is a $1$-Lipschitz image of $Z$ under $T_{t}=(1-t)T_{0}+tT_{1}$. Given that $Z$ satisfies a Poincaré inequality with constant $1$, a standard change of variables argument immediately shows that $X_{t}$ satisfies the Poincaré inequality with constant $1$ for all functions. However, interestingly, we do not use this fact directly. Instead, we use the $1$-Lipschitz property of $T_{t}$ and the Poincaré constant of $Z$ separately. It remains an open question whether the optimal coupling also satisfies the conclusion of Theorem 1.5.

An important feature of the interpolation $X_{t}$, if considered under optimal coupling, is that the trajectories $\{T_{t}(x)\}_{t\in(0,1)}$ do not cross (in an almost-everywhere sense), and hence the distribution $\mu_{t}$ of $X_{t}$ can be described as the flow of $\mu_{0}$ under a time-dependent velocity field. Yet another useful property under optimal coupling is that the velocity field generated is a gradient field. Both these properties are used in [1].

In our case, for $X_{t}$ that we consider, we are not guaranteed the existence of a driving velocity field, nor do we see a reason for this velocity field to be a gradient field even if it exists. The former technical difficulty is overcome by a “trajectories do not cross” result when $X_{0},X_{1}$ are “nice” (Proposition 2.2), and approximation. The latter issue most prominently appears in the proof of Theorem 1.5, where an inequality such as $\mathbb{E}\textnormal{tr}[\nabla v(X_{t})^{2}]\geq\mathbb{E}|v(X_{t})|^{2}$ is needed for a particular odd vector field $v$. This is always true when $v$ is a gradient field and the even random vector $X_{t}$ has Poincaré constant $1$, but not in general. To resolve this problem, we explicitly use the structure of the given vector field $v$ (which depends on $T_{t}$) and the Gaussian Poincaré inequality. This makes it unclear if our proof would go through if $T_{0}$ and $T_{1}$ were contractions (via the reverse Ornstein–Uhlenbeck process) introduced by Kim and E. Milman [20], and not Brenier maps. Readers familiar with the work of Alesker, Gilboa, and V. Milman [2] may find it intriguing to compare the fact that the coupling used in this paper admits a Theorem 1.5 (while for other aforementioned couplings such a result is yet unestablished), with Gromov’s observation that $\nabla\phi[\mathbb{R}^{n}]+\nabla\psi[\mathbb{R}^{n}]=(\nabla\phi+\nabla\psi)[\mathbb{R}^{n}]$ when $\phi,\psi$ are $C^{2}$ convex functions with strictly positive Hessian [19, 1.3.A.] (see also, [2, Proposition 2.2]).

As an immediate corollary to Theorem 1.5, using the variational principle (7), we obtain a new proof of Eskenazis and Moschidis’ result.

A fundamental problem in this area concerns obtaining functional forms of geometric inequalities. This means, given a geometric inequality, one wants to find a functional inequality which recovers the given geometric inequality when applied to functions canonically associated with the involved sets (for example, to indicator functions). For a prototypical example, consider the Brunn–Minkowski inequality in its geometric-mean form that all log-concave measures $\nu$ are known to satisfy:

whenever $K_{0}$ and $K_{1}$ are compact sets in $\mathbb{R}^{n}$. The functional form of (12) is the Prékopa–Leindler inequality which concludes

whenever $f,g,h$ are non-negative functions satisfying

for all $x,y$. Of course, if $f$ and $g$ are indicator functions of $K_{0}$ and $K_{1}$, respectively, then the indicator of $(1-t)K_{0}+tK_{1}$ is an admissible choice for $h$, thus producing (12). While several proofs of the Prékopa–Leindler inequality exist (for example, see [16]), an elegant proof can be obtained from the entropy form of (12): every pair of $\mathbb{R}^{n}$-valued random vectors $X_{0},X_{1}$ with density (with respect to the Lebesgue measure) have a joint distribution $(X_{0},X_{1})$ such that,

where $Y$ is a random vector with distribution $\nu$. The fact that the optimal coupling $(X_{0},X_{1})$ satisfies (15) (see [3, Theorem 9.4.11]) is well known, and often recorded as the “displacement convexity of entropy on the metric measure space $(\mathbb{R}^{n},|\cdot|,\nu)$”. To go from (15) to (13) one can use the Donsker–Varadhan duality formula [13, Section 2] describing the Legendre transform of relative entropy. It says, for $\nu$-integrable functions $\phi$, we have

where the supremum on the right is over all probability measures $\mu$ absolutely continuous with respect to $\nu$, and equality is attained in (16) for $\textnormal{ d}\mu\propto e^{\phi}\textnormal{ d}\nu$.

We do not spell out the details of the implication (15) $\Rightarrow$ (13) because the reader may infer the general idea from our proof of Theorem 1.7 which is rather short. Nonetheless, it is apt to remark here that this technique stands out because it entirely operates at the level of integrals and does not appeal to local estimates on the integrands (other than the one granted by assumption), thereby making it possible to extract functional inequalities even if a convexity property of entropy is only available on a restricted class of measures. The same cannot be said about some other transport-based proofs of the Prékopa–Leindler inequality (or its generalisations). Besides, this method works in measure spaces without any smooth structure. For example, the reader may find beautiful applications to discrete structures in works of Gozlan, Roberto, Samson, and Tetali [18], and Slomka [28].

We will use this duality to obtain a functional form of the dimensional Brunn–Minkowski inequality (3), which is our second main result.

Indeed, when $f$ and $g$ are indicators of symmetric convex bodies, and $p=\infty$, one recovers (3). Inequalities such as in the theorem above are sometimes called Borell–Brascamp–Lieb inequalities after the works by Borell [5], and Brascamp–Lieb [7].

To the best of our knowledge, the argument for obtaining Theorem 1.7 from Theorem 1.5, though simple, is new. Previously, it was not clear how to apply duality (16) to inequalities such as (9) that are not linear in relative entropy. Exactly the same idea as we use in the proof of Theorem 1.7 gives further dimension-dependent functional inequalities for functions that are not necessarily log-concave, as discussed below.

Recall that, a convex function $V:\mathbb{R}^{n}\to\mathbb{R}$ is said to be $\beta$-homogeneous if $V(\lambda x)=\lambda^{\beta}V(x)$, for every $x\in\mathbb{R}^{n}$ and $\lambda>0$. Consider the probability measure $\nu\propto e^{-V}\textnormal{ d}x$, such that $V$ is $\beta$-homogeneous for some $\beta\in(1,\infty)$. Say $\nu$ is represented by a random vector $Y$. Then, [1, Theorem 1.4] states that, for random vectors $X_{0}$ and $X_{1}$ having radially decreasing density with respect to $\nu$, there exists a coupling $(X_{0},X_{1})$ (in fact, the optimal coupling works) such that

From this, we have the following result.

The standard Gaussian measure falls under the regime $\beta=2$. Thus, Theorem 1.8 can be applied to a larger class of functions compared to Theorem 1.7, but at the same time draws a weaker conclusion.

First of all, we note that $\frac{\textnormal{ d}^{2}}{\textnormal{ d}t^{2}}e^{-\frac{1}{n}D(X_{t}\|Z)}\leq 0$ is equivalent to $\frac{\textnormal{ d}^{2}}{\textnormal{ d}t^{2}}D(X_{t}\|Z)\geq\frac{1}{n}\left(\frac{\textnormal{ d}}{\textnormal{ d}t}D(X_{t}\|Z)\right)^{2}$, whenever the relevant quantities have the required regularity. Thus, we would like to compute $\frac{\textnormal{ d}^{2}}{\textnormal{ d}t^{2}}D(X_{t}\|Z)$ and $\frac{\textnormal{ d}}{\textnormal{ d}t}D(X_{t}\|Z)$. Such local computations are often best performed in the language of velocity fields.

Suppose a curve $\{\mu_{t}\}_{t\in[0,1]}$ of probability measures on $\mathbb{R}^{n}$ is given. A time-dependent velocity field $v_{t}$ is said to be compatible with $\{\mu_{t}\}_{t\in[0,1]}$ if

is satisfied in the weak sense, where div denotes divergence. The latter equation means that

for all compactly supported smooth functions $f$. Once Equation (23) is known to hold, Equation (24) holds under wider generality; for example, it holds for all bounded Lipschitz smooth functions (see [3, Chapter 8]).

Ignoring all regularity issues, we compute the derivatives of $D(\mu_{t}\|\gamma)$ twice when a compatible velocity field is given. We write the result for a general log-concave measure $\nu$ since it may be of independent interest.

To utilise the above formulas for the derivatives of entropy, we need to establish the existence of compatible velocity fields in the cases of interest. We let $I_{n\times n}$ denote the $n\times n$ identity matrix.

In this section, we solely work with the Gaussian measure as the reference measure. Thus, $\nu$ from the previous section is taken to be $\gamma$. In this case, we will denote $\textnormal{div}^{W}$ by $\widetilde{\textnormal{div}}$ and $\mathcal{G}^{W}$ by $\widetilde{\mathcal{G}}$.