1 Introduction↩︎

Our investigation begins with the celebrated Prékopa–Leindler inequality [1], [2]. Consider three integrable functions \(f,g,h: \mathbb{R}^n \to [0,\infty)\) such that, for every \(t \in [0,1]\), \[h((1-t)x+ty) \geq f(x)^{(1-t)}g(y)^t \quad \text{ for every } x,y \in \mathbb{R}^n.\] Then, \[\label{eq:PL951} \int_{\mathbb{R}^n} h \, \mathrm{d}x \geq \left( \int_{\mathbb{R}^n} f \, \mathrm{d}x \right)^{(1-t)} \left( \int_{\mathbb{R}^n} g \, \mathrm{d}x \right)^t \quad \text{ for every } t \in [0,1],\tag{1}\] where integration in \(\, \mathrm{d}x\) is understood with respect to the Lebesgue measure. This inequality is deeply connected to the Brunn–Minkowski inequality, of which it can be considered the functional analogue. See Gardner’s survey [3] for more details on these two inequalities, and Schneider’s monograph [4] for a thorough introduction to the Brunn–Minkowski theory.

The first immediate corollary of 1 is that, thanks to a well-known classification result by Borell [5], a measure \(\mu\) on \(\mathbb{R}^n\) such that \(\, \mathrm{d}\mu(x)=e^{-u(x)}\, \mathrm{d}x\), where \(u:\mathbb{R}^n \to \mathbb{R}\cup \{\infty\}\) is a lower semi-continuous convex function, is log-concave. That is, \[\label{eq:log95concave95measure} \mu((1-t)K+tL) \geq \mu(K)^{(1-t)}\mu(L)^t\tag{2}\] for every non-empty and compact convex sets \(K,L \in \mathbb{R}^n\) and \(t \in [0,1]\). The set-operation \(+\) denotes the usual Minkowski addition (see, for example, [4]). If \(\mu\) is chosen to be the Lebesgue measure on \(\mathbb{R}^n\), 2 is an equivalent form of the Brunn–Minkowski inequality.

A deeper consequence of the log-concavity property 2 is the validity of certain Poincaré-type inequalities on the boundaries of convex sets. As before, suppose that \(\, \mathrm{d}\mu(x)=e^{-u(x)}\, \mathrm{d}x\) where \(u:\mathbb{R}^n \to \mathbb{R}\cup \{\infty\}\) is a lower semi-continuous convex function. Assume in addition that \(u\in C^1(\mathbb{R}^n)\) and consider a compact convex set \(K \subset \mathbb{R}^n\) whose boundary \(\partial K\) is a strictly convex \(C^2\) hypersurface. We shall say, with a slight abuse of terminology, that \(\partial K\) is strictly convex whenever its second fundamental form \({\rm II}\) is positive definite. Then, for every locally Lipschitz function \(\rho:\partial K\to\mathbb{R}\), the following inequality holds: \[\label{eq:Poincare} \int_{\partial K} {\rm H}_\mu \rho^2 \, \mathrm{d}\mu-\frac{1}{\mu(K)}\left(\int_{\partial K} \rho \, \mathrm{d}\mu\right)^2\leq\int_{\partial K} \langle {\rm II}^{-1} \nabla_{\partial K} \rho, \nabla_{\partial K} \rho \rangle \, \mathrm{d}\mu .\tag{3}\] Here, \(\nabla_{\partial K}\) denotes the tangential gradient, and \(\langle\cdot,\cdot\rangle\) the restriction of the Euclidean scalar product to the tangent bundle of \(\partial K\). With a slight abuse of notation, we use the same symbol for the scalar product on the whole space. The weighted mean curvature is given by \({\rm H}_\mu=\mathrm{tr}({\rm II})-\langle \nabla u,\nu_K\rangle\), where \(\nu_K\) denotes the outer unit normal to \(\partial K\). Integration on \(\partial K\) with respect to \(\mu\) is understood as integration against the \((n-1)\)-dimensional Hausdorff measure weighted by \(e^{-u}\). An equivalent form of 3 was proved by Colesanti [6] in the case of the Lebesgue measure, and extended to a much more general setting by Kolesnikov and Milman [7].

Another consequence of 1 we are interested in is the eponymous variance inequality proved by Brascamp and Lieb in [8], which we later refer to as Brascamp–Lieb inequality, for short. Consider \(u \in C^2(\mathbb{R}^n)\) such that the Hessian matrix of \(u\), which we denote by \(\nabla^2 u\), is positive definite. Then, for every locally Lipschitz function \(\varphi: \mathbb{R}^n \to \mathbb{R}\) and compact convex set \(K \subset \mathbb{R}^n\) with non-empty interior, the inequality \[\label{eq:Brascamp95Lieb} \int_K \varphi^2 \, \mathrm{d}\mu - \frac{1}{\mu(K)}\left( \int_K \varphi \, \mathrm{d}\mu\right)^2 \leq \int_K \langle (\nabla^2 u)^{-1} \nabla \varphi, \nabla \varphi\rangle \, \mathrm{d}\mu\tag{4}\] holds. A far-reaching generalization of 4 was obtained by Kolesnikov and Milman in [9].

Both 3 and 4 can be obtained from 1 as infinitesimal forms. This process can be summarized as follows: Consider a function \(\Phi: \mathbb{R}^n \times [0,1] \to \mathbb{R}\cup \{ \infty\}\) such that it is jointly convex. Then, 1 implies that \[\mathcal{S}(t)=\log \left(\int_{\mathbb{R}^n} e^{-\Phi(t,x)} \, \mathrm{d}x \right) \quad \text{ is concave for } t \in [0,1].\] Statements of this kind are known as concavity principles for marginals. For more details on the topic, see, for example, the recent work of Cordero-Erausquin and Eskenazis [10], which was of great inspiration in the preparation of this manuscript. If one proves that \(\mathcal{S}\) is twice differentiable as \(t \to 0^+\), the concavity of \(\mathcal{S}\) can be equivalently stated as \(\mathcal{S}''(0)\leq 0\). Writing explicitly \(\mathcal{S}''(0)\) gives, for appropriate choices of \(\Phi\), the inequalities 3 and 4 .

In this work, after deducing a new infinitesimal form for 1 , we will apply this machinery to obtain a stability estimate for 3 . In turn, with the tools thus developed, we will show a series of applications in the direction of the dimensional Brunn–Minkowski conjecture (see later 10 ).

1.0.0.1 A new infinitesimal form of the Prékopa–Leindler inequality.

We have summarized above how the Prékopa–Leindler inequality can be differentiated to obtain two kinds of Poincaré-type inequalities. The first contribution of this paper is showing that 3 and 4 appear simultaneously (together with some new terms!) by differentiation of an appropriate choice of \(\Phi\) in \(\mathcal{S}\). Before stating this result, we introduce some new notation. Consider two Lipschitz functions \(\rho_0, \rho_1 : \partial K \to \mathbb{R}\). Then, for \(K\) and \(u\) fixed and satisfying the previous hypotheses, 3 implies that the symmetric bilinear form \[\label{eq:P} \langle \rho_0,\rho_1 \rangle_{\rm P}= \int_{\partial K} \langle {\rm II}^{-1} \nabla_{\partial K} \rho_0, \nabla_{\partial K} \rho_1 \rangle \, \mathrm{d}\mu - \int_{\partial K} {\rm H}_\mu \rho_0 \rho_1 \, \mathrm{d}\mu + \frac{1}{\mu(K)}\left(\int_{\partial K} \rho_0 \, \mathrm{d}\mu\right)\left(\int_{\partial K} \rho_1 \, \mathrm{d}\mu\right) \tag{5}\] is positive semi-definite. Similarly, for two locally Lipschitz functions \(\varphi_0, \varphi_1: \mathbb{R}^n \to \mathbb{R}\), 4 implies that the symmetric bilinear form \[\label{eq:BL} \langle \varphi_0, \varphi_1 \rangle_{\rm BL}= \int_K \langle (\nabla^2 u)^{-1} \nabla \varphi_0, \nabla \varphi_1\rangle \, \mathrm{d}\mu - \int_K \varphi_0 \varphi_1 \, \mathrm{d}\mu + \frac{1}{\mu(K)}\left(\int_K \varphi_0 \, \mathrm{d}\mu \right)\left( \int_K \varphi_1 \, \mathrm{d}\mu \right) \tag{6}\] is positive semi-definite.

Before presenting our first main result, we need a further bilinear form. Consider a Lipschitz function \(\rho: \partial K \to \mathbb{R}\) and a locally Lipschitz function \(\varphi: \mathbb{R}^n \to \mathbb{R}\). Under the regularity assumptions on \(u\) and \(K\) needed for 3 and 4 , we define \[\label{eq:I} \langle \rho, \varphi \rangle_{\rm I} = \int_{\partial K} \rho \varphi \, \mathrm{d}\mu - \frac{1}{\mu(K)}\left( \int_{\partial K} \rho \, \mathrm{d}\mu \right) \left( \int_K \varphi \, \mathrm{d}\mu \right), \tag{7}\] which can be understood as an interaction term between 3 and 4 (justifying the choice of the subscript). Our first result, which takes the form of a Cauchy–Schwarz-type inequality, reads as follows.

We will infer 8 from another statement (see Theorem 7) which, under the same assumptions of Theorem 1, reads as \[\langle \rho, \varphi \rangle_{\rm I} \leq \frac{\langle \rho, \rho \rangle_{\rm P} + \langle \varphi, \varphi \rangle_{\rm BL}}{2}.\] From this statement it is immediate to derive 3 and 4 choosing \(\varphi \equiv 0\) and \(\rho \equiv 0\), respectively. The inequality 8 , instead, does not look like a direct generalization of 3 and 4 . However, it yields a sharper lower bound. Indeed, once we know that \(\langle \cdot ,\cdot \rangle_{\rm P} \geq 0\) we obtain, for example, \(\langle\varphi,\varphi\rangle_{\rm BL} \geq \langle \rho,\varphi\rangle_{\rm I}^2/\langle \rho,\rho\rangle_{\rm P}\) for every choice of \(\rho\) such that \(\langle \rho,\rho\rangle _{\rm P} \neq 0\), which improves the non-negativity implied by 4 . We shall see in the sequel that any \(\rho\not\equiv 0\) does satisfy \(\langle \rho,\rho \rangle_{\rm P} >0\). That is, \(\langle \cdot , \cdot \rangle_{\rm P}\) is positive definite. On the other hand, Livshyts [11] showed that when the convex set \(K\) in 4 is compact, \(\langle \varphi, \varphi \rangle_{\rm BL} =0\) if and only if \(\varphi\) is constant, and we may infer a similar improvement of the Poincaré inequality testing 8 against any non-constant function \(\varphi\). We do not investigate the equality cases of 8 . Nevertheless, in Proposition 8 we show that this new inequality admits additional equality cases beyond those implied by 3 and 4 .

The proof of Theorem 1 is based on a perturbative argument developed by the second-named author in [12], exploiting the connection between Wulff shapes of convex bodies and the Fenchel–Legendre transform of convex functions. The perturbations thus obtained allow the simultaneous treatment of 3 and 4 , highlighting the similarities between the two inequalities. This method has already found applications to geometric inequalities in [13], by Mussnig and the second-named author.

1.0.0.2 Stability estimates and the dimensional Brunn–Minkowski conjecture.

After obtaining Theorem 1, in the remainder of this manuscript, we focus on a series of applications. First, we shall prove a stability estimate for 3 . In turn, the results thus obtained will provide the background for some surprising applications to the dimensional Brunn–Minkowski conjecture. Before proceeding further with the exposition, we report some brief historical context for the topic. Further details can be found in the references in the following paragraph.

The Poincaré inequality 3 and the Brascamp–Lieb inequality 4 , together with suitable variants, have been the subject of intensive research in the last couple of decades. As mentioned above, Colesanti [6] obtained a non-weighted version of 3 . In the same period, Bobkov and Ledoux [14], [15] generalized 4 to \(\beta\)-concave functions computing an infinitesimal form of the Borell–Brascamp–Lieb inequality (compare [5], [8]), a generalization of the Prékopa–Leindler inequality 1 . We point out that our methods can be easily extended to obtain a generalization of Theorem 1 stemming from the Borell–Brascamp–Lieb inequality. Nonetheless, we decided to focus on the log-concave case for a cleaner and more accessible exposition. Around the same time, Cordero-Erausquin, Fradelizi, and Maurey [16] used 4 to prove the (B)-conjecture for the Gaussian measure, pioneering the so-called \(L_2\) method in the context of geometric and analytic inequalities. Almost a decade later, Kolesnikov and Milman [7], [9] extended this approach to the setting of weighted Riemannian manifolds under suitable curvature assumptions, reigniting interest in the topic. With similar techniques, Nguyen [17] extended the results of [14]. Afterwards, the application of the \(L_2\) method has found several applications in the extension of the (B)-conjecture to even log-concave probability measures [10], [18], [19] (which we will not treat in this work) and to the dimensional Brunn–Minkowski inequality [10], [19]–[23] (which we present later in 10 ). Both conjectures, already interesting on their own, have acquired further relevance in relation to the logarithmic Brunn–Minkowski conjecture, stated in [24] by Böröczky, Lutwak, Yang, and Zhang, and are intimately connected to the logarithmic Minkowski problem, introduced by the same authors in [25]. Thanks to the works of Saroglou [26] and Livshyts, Marsiglietti, Nayar, and Zvavitch [27], it is known that the logarithmic Brunn–Minkowski conjecture implies the dimensional Brunn–Minkowski conjecture and the (B)-conjecture. Further details on the development of these open problems can be found in [10].

Given a compact convex set \(K\subset\mathbb{R}^n\) with non-empty interior and a measure \(\mu\) with sufficiently regular density with respect to the Lebesgue measure, we denote by \(H^1(\partial K,\mu)\) the weighted Sobolev space of real-valued functions on \(\partial K\) with respect to \(\mu\). The associated Sobolev norm is denoted by \(\|\cdot\|_{H^1(\partial K,\mu)}\). Similarly, one defines \(H^1(K,\mu)\) and the corresponding norm; see Section 3 for precise definitions. These spaces provide a natural extension for the choice of \(\rho\) and \(\varphi\) in Theorem 1, where the two functions appear in a completely decoupled manner. Extending the validity of 8 to the Sobolev spaces above justifies treating \(\varphi\) as a parameter, allowing one to derive sharper inequalities for a fixed function \(\rho\in H^1(\partial K,\mu)\). By exploiting a suitable Sobolev extension of \(\rho\) to the whole set \(K\), we obtain an interpolation-type inequality (see Proposition 9) which, in turn, yields the following stability result for the weighted Poincaré inequality.

Let us now shortly present the dimensional Brunn–Minkowski conjecture. We say that \(A \subset \mathbb{R}^n\) is origin-symmetric if \(A=-A=\{-x \colon x \in A \}\), and that a measure \(\mu\) is even if \(\mu(B)=\mu(-B)\) for every \(\mu\)-measurable set \(B \subset \mathbb{R}^n\). The question is the following: Given an even log-concave measure \(\mu\) on \(\mathbb{R}^n\), is it true that, for every pair of origin-symmetric non-empty compact convex sets \(K,L \subset \mathbb{R}^n\), the inequality \[\label{eq:dim95BM} \mu((1-t)K+tL)^{\frac{1}{n}}\geq (1-t)\mu(K)^{\frac{1}{n}}+t\mu(L)^{\frac{1}{n}}\tag{10}\] holds for every \(t \in [0,1]\)? In other words, does every even log-concave measure enjoy the same dimensional concavity as the Lebesgue measure when restricted to the class of origin-symmetric convex sets?

This conjecture was put forth by Gardner and Zvavitch for the Gaussian measure [28] and was resolved by Eskenazis and Moschidis [20]. Cordero-Erausquin and Rotem [19] gave an affirmative answer for sufficiently regular and rotationally invariant log-concave measures, while Kolesnikov and Livshyts [22] established the conjecture for hereditarily convex log-concave measures. For general even log-concave measures, Livshyts [23] proved 10 with exponent \(n^{-(4+o(1))}\), which currently constitutes the best known universal bound.

To study the dimensional Brunn–Minkowski conjecture, Livshyts [11] proposed the following local approach. Given a log-concave measure \(\mu\) on \(\mathbb{R}^n\) and a compact convex set \(K \subset \mathbb{R}^n\) with non-empty interior, one seeks the maximal value \(p(\mu,K)\) such that \[\left. \frac{\, \mathrm{d}^2}{\, \mathrm{d}t^2} \mu\bigl(K(\rho,t)\bigr)^{p(\mu,K)} \right|_{t=0}\leq 0\quad \text{for every } \rho \in \mathcal{A},\] where \(K(\rho,t)\) denotes a perturbation of \(K\) generated by a suitable function \(\rho : \partial K \to \mathbb{R}\), and \(\mathcal{A}\) is the class of admissible perturbations (see Section 4 for precise definitions). Building upon this idea, we shall reformulate the dimensional Brunn–Minkowski conjecture in a variational fashion, where the energy functional to be minimized turns out to depend crucially on the bilinear form \(\langle \cdot, \cdot \rangle_{\rm P}\). With the main theorems we mentioned so far, as well as some preliminary estimates, we prove that not only is 3 saturated by trivial functions, but that \(\langle \cdot, \cdot \rangle_{\rm P}\) is in addition coercive, therefore allowing the application of the Lax–Milgram theorem to \(\langle \cdot, \cdot \rangle_{\rm P}\). From this procedure, we deduce the following result, which we formulate here as a statement on the existence and uniqueness of a weak solution to the corresponding Euler–Lagrange equation. Given a vector field \(X \in C^{1}(\partial K,\mathbb{R}^n)\), we denote by \(\nabla_{\partial K} \cdot X\) the tangential divergence of \(X\) on \(\partial K\).

It is worth mentioning that the operator \(-\nabla_{\partial K} \cdot ({\rm II}^{-1}\nabla_{\partial K} (\cdot) )+\langle \nabla_{\partial K}u,{\rm II}^{-1}\nabla_{\partial K}(\cdot) \rangle\) is nothing but a weighted Laplacian on \(\partial K\) with an appropriate choice of metric and measure. Compare, e.g., [7].

This equation paves the way to several reformulations of the conjecture 10 . For the moment, we highlight the following consequence (see Theorem 13 later), which relies on the bilinear form \(\langle \cdot, \cdot \rangle_{\rm I}\) introduced earlier. In particular, the dimensional Brunn–Minkowski conjecture is equivalent to the following local statement: For every origin-symmetric \(K\) and even \(u\) satisfying the assumptions of Theorem 3, the inequality \[\label{eq:equivalent95form95intro} \langle \overline{\rho}, \langle \nabla u, x \rangle \rangle_{\rm I}+\int_K \langle \nabla u, x \rangle\, \, \mathrm{d}\mu \geq 0\tag{13}\] holds, where \(\overline{\rho}\) is the function provided by Theorem 3. Indeed, \(\overline{\rho}\) may be interpreted as an extremal perturbation of \(K\), encoding the concavity properties of \(K\) while simultaneously incorporating the geometry induced by the measure \(\mu\). It is interesting to note that in the presence of symmetry, the minimizer is even. Moreover, we note that estimates similar to 13 were already required in [10] (see Lemma 11 therein) to establish both the (B)-conjecture and the dimensional Brunn–Minkowski conjecture, together with certain spectral inequalities. By contrast, the Euler–Lagrange approach adopted here appears to bypass the need for such spectral inequalities, at the cost of a more delicate analysis of the PDE 11 .

We conclude this manuscript with a result in the same spirit as [22], due to Kolesnikov and Livshyts, obtained under the same pinching conditions on the measure \(\mu\). We denote the identity matrix by \(\mathrm{Id}\).

Notice that the estimate obtained in [22] is slightly sharper (even though morally equivalent up to an absolute constant). In the notation above, it corresponds to the constant \(c=2/(\sqrt{r}+1)^2\). The bound derived here, however, is obtained through a substantially different procedure, building directly on the framework developed in the preceding sections. In particular, we believe this different method to be of independent interest. Finally, we note that, when \(\mu\) is the Gaussian measure, Theorem 4 implies the main result of [22]; in particular, a close inspection of the proof implies that the symmetry assumption on the convex sets involved can be weakened to the assumption that the sets merely contain the origin.

1.0.0.3 Structure of the paper.

We start in Section 2 with an exposition of the perturbative methods necessary for our computations. In the same section, we prove Theorem 1. In Section 3, we extend the bilinear forms \(\langle\cdot,\cdot \rangle_{\rm P}\) and \(\langle\cdot,\cdot \rangle_{\rm BL}\) to suitable Sobolev spaces, providing the machinery needed for Theorem 2, and the proof of the theorem itself. We then start Section 4 proving that \(\langle\cdot,\cdot \rangle_{\rm P}\) satisfies the assumptions of the Lax–Milgram theorem, which will imply Theorem 3. In the same section, we discuss the corresponding Euler–Lagrange equation and some reformulations of the dimensional Brunn–Minkowski conjecture. The manuscript is concluded in Section 5 with the proof of Theorem 4.

2 A concavity principle along functional Wulff shapes↩︎

For the convenience of the reader, we report first some preliminaries on convex sets and convex functions. For some complete introductions on these topics, the reader can consult the monographs by Schneider [4] and Rockafellar [29]. For \(A, B \subset \mathbb{R}^n\), their Minkowski sum is the set \[A+B=\{ x+y \in \mathbb{R}^n: x\in A, y \in B\}.\] We denote by \(\mathcal{K}^n\) the family of compact convex subsets of \(\mathbb{R}^n\) with non-empty interior, also known as convex bodies. This space is closed under Minkowski addition. If \(K \in \mathcal{K}^n\), then it is uniquely determined by its support function \(h_K(x)=\sup_{y \in K} \langle x, y \rangle, x \in \mathbb{R}^n\). By construction, support functions are positively \(1\)-homogeneous, and can thus be identified with their restrictions to the Euclidean unit sphere \(\mathbb{S}^{n-1}\) of \(\mathbb{R}^n\). A convex body \(K \in \mathcal{K}^n\) can be represented via its support function as \[K=\{x \in \mathbb{R}^n : \langle x,\xi \rangle \leq h_K(\xi) \text{ for every }\xi \in \mathbb{S}^{n-1} \}.\] This representation is connected with the so-called Wulff shape construction. Consider \(f \in C(\mathbb{S}^{n-1})\). Then, the corresponding Wulff shape is the set \[[f]=\{x \in \mathbb{R}^n: \langle x , \xi \rangle \leq f(\xi) \text{ for every }\xi \in \mathbb{S}^{n-1} \}.\] It is immediate to prove that the set thus obtained is always convex. Moreover, \([f] \in \mathcal{K}^n\) whenever \(f>0\) up to the addition of a linear function.

Consider now the space of convex functions \[\mathrm{Conv}(\mathbb{R}^n)=\{u: \mathbb{R}^n \to \mathbb{R}\cup \{\infty\}: u \text{ is convex, lower semi-continuous, and proper} \},\] where a function \(u\) is proper if it is not identically equal to \(\infty\). Equivalently, \(\mathrm{Conv}(\mathbb{R}^n)\) is the space of functions \(u: \mathbb{R}^n \to \mathbb{R}\cup\infty\) such that the corresponding epigraph \(\mathrm{epi}(u)=\{(x,z) \in \mathbb{R}^n \times \mathbb{R}: u(x) \leq z \}\) is convex, closed, and non-empty. On \(\mathrm{Conv}(\mathbb{R}^n)\), there is an operation which is equivalent to the Minkowski addition. Given \(u, v \in \mathrm{Conv}(\mathbb{R}^n)\), their infimal convolution is the function \(u \square v\), which is characterized by the property \(\mathrm{epi}(u \square v)=\mathrm{epi}(u)+\mathrm{epi}(v)\). Thus, \(u \square v \in \mathrm{Conv}(\mathbb{R}^n)\). Alternatively, this operation can be explicitly written as \[u \square v(x)= \inf_{y \in \mathbb{R}^n} \{ u(y)+v(x-y) \}.\] A further operation on the set of convex functions is the one of epi-multiplication. For \(u \in \mathrm{Conv}(\mathbb{R}^n)\) and \(t>0\), the function \(t \mathbin{\vcenter{\rule{.3ex}{.3ex}}}u\) is described by the condition \(\mathrm{epi}(t \mathbin{\vcenter{\rule{.3ex}{.3ex}}}u)=t\mathrm{epi}(u)\). Equivalently, \[t\mathbin{\vcenter{\rule{.3ex}{.3ex}}}u(x) = t u\left( \frac{x}{t} \right).\] Observe that \(\mathcal{K}^n\) can be identified inside \(\mathrm{Conv}(\mathbb{R}^n)\) as the subset of indicator functions. That is, for \(K \in \mathcal{K}^n\) we consider \(I_K \in \mathrm{Conv}(\mathbb{R}^n)\) defined as \[\begin{align} I_K(x)=\begin{cases} 0 & \text{ if }x \in \mathcal{K},\\ \infty & \text{ otherwise.} \end{cases} \end{align}\] Since \(h_K \in \mathrm{Conv}(\mathbb{R}^n)\) for every \(K \in \mathcal{K}^n\), support functions provide a further representation of convex bodies inside \(\mathrm{Conv}(\mathbb{R}^n)\). We say that a function on \(\mathbb{S}^{n-1}\) is convex if its natural positive \(1\)-homogeneous extension is convex. Observe that if \(f \in C(\mathbb{S}^{n-1})\) is convex, then \(f=h_{[f]}\).

We now introduce a fundamental tool for this section. Given a function \(\psi: \mathbb{R}^n \to \mathbb{R}\cup \{\infty\}\), its Fenchel–Legendre transform is \[\psi^*(x)=\sup_{y \in \mathbb{R}^n}\{ \langle x, y \rangle - \psi(y)\}.\] It is well known that if \(u \in \mathrm{Conv}(\mathbb{R}^n)\) then \(u^* \in \mathrm{Conv}(\mathbb{R}^n)\) and \((u^*)^*=u\). We will use several well-known properties of this transform. For more details, the reader can consult [29]. We recall, in particular, that for \(f \in C(\mathbb{S}^{n-1})\) one has the identity \[\label{eq:FL95Wulff} f^*=I_{[f]},\tag{14}\] where \(f\) is identified with its \(1\)-homogeneous extension. As a special case, observe that \(h_K^*=I_K\) for every \(K \in \mathcal{K}^n\). Another useful property of the Fenchel–Legendre transform is the following: For \(u,v \in \mathrm{Conv}(\mathbb{R}^n)\) and \(t,s>0\), \[\label{eq:sums95FL} (tu^*+sv^*)^*=(t \mathbin{\vcenter{\rule{.3ex}{.3ex}}}u) \square (s \mathbin{\vcenter{\rule{.3ex}{.3ex}}}v).\tag{15}\] In [12], the second-named author investigated the connections between Wulff shapes and the Fenchel–Legendre transform. In particular, the latter can be regarded as a functional version of the former. We will largely use this intuition in the remainder of this section.

Theorem 1 will be obtained by differentiating the following concavity principle, which is a direct application of [2].

As we will soon have to compute the first and second derivatives of 16 , we need to assume further regularity on the sets and functions involved. We say that \(K \in \mathcal{K}^n\) is a smooth convex body if \(\partial K\) is a manifold of class \(C^2\) and strictly convex in the sense described in the introduction, i.e., \({\rm II} > 0\). In particular, \(h_K\) is of class \(C^2\) in \(\mathbb{R}^n \setminus \{ o\}\) (compare [4]) and the main curvatures of \(\partial K\) are strictly positive and bounded as functions on \(\partial K\). Additionally, in this section, we will always tacitly assume that \(o \in \mathrm{int}K\) (the interior of \(K\)), as the problems at hand will always be invariant under translation. We say that a function \(u \in \mathrm{Conv}(\mathbb{R}^n)\) is smooth and strictly convex if \(u \in C^2(\mathbb{R}^n)\) and \(\nabla^2 u\) is positive definite. We now provide a statement concerning the computation of shape derivatives. In the context we are treating, similar facts have been proved in [7] and [10]. For a function \(\Phi_t\) differentiable in \(t\) (and which might depend on other variables), we adopt the notation \(\Phi'_t=\frac{\partial }{\partial t} \Phi_t\) for the partial derivative on the entry of \(t\) only.

We now introduce our choice for \(X_t\), which can be compared with the flow studied in [7]. Consider a smooth convex body \(K \subset \mathbb{R}^n\) such that \(o \in {\rm int} K\) and \(f \in C^2(\mathbb{S}^{n-1})\). Then, for \(t \in (-\varepsilon,\varepsilon)\) for \(\varepsilon>0\) sufficiently small, the Wulff shape \(K_t=[h_K+tf]\) is a smooth convex body such that \(h_{K_t}=h_K+tf\) and \(o \in {\rm int}K_t\). We first define \(X_t\) on \(\partial K\) as \[\begin{align} X_t : \partial K &\to \partial K_t \\ x &\mapsto \nu^{-1}_{K_t}\circ \nu_K(x), \end{align}\] where \(t \in (-\varepsilon,\varepsilon)\). By a well-known property of support functions (compare, e.g., [4]) and by the representation of \(h_{K_t}\), for every \(\xi \in \mathbb{S}^{n-1}\), \[\nu^{-1}_{K_t}(\xi)= \nabla h_{K_t}(\xi)=\nabla (h_K+tf)=\nu^{-1}_K(\xi)+t\nabla f(\xi).\] Choosing \(\xi=\nu_K(x)\) for \(x \in \partial K,\) we infer \[X_t(x)=x+t \nabla f(\nu_K(x)).\] With the next lemma, we extend \(X_t\) to \(\mathbb{R}^n\). For \(K \in \mathcal{K}^n\) such that \(o \in {\rm int}K\) we denote by \[||x||_K=\inf\{\tau \geq 0 : x \in \tau K \}, \, x \in \mathbb{R}^n,\] its gauge function (compare [4]).

The next step is to establish hypotheses under which the integrand in 16 is sufficiently regular to apply Proposition 5. Consider a smooth and strictly convex function \(u\). Additionally, assume that there exist \(k_2 \geq k_1 >0\) such that \(k_1 \mathrm{Id}\leq \nabla^2 u \leq k_2 \mathrm{Id}\). Equivalently, all the eigenvalues of \(\nabla^2 u\) lie between \(k_1\) and \(k_2\). By the properties of the Fenchel–Legendre transform, this implies that \(k_2^{-1} \mathrm{Id}\leq \nabla^2 u^* \leq k_1^{-1} \mathrm{Id}\). Therefore, if we choose \(\psi \in C^2(\mathbb{R}^n)\) such that its second derivatives are uniformly bounded on \(\mathbb{R}^n\), the functions \(u^*+t\psi\) and \((u^*+t\psi)^*\) are both smooth and strictly convex for \(t\) sufficiently small. We recall the following result, which can be considered folklore under our regularity assumptions. See, for example, [32].

We now have all the ingredients for explicitly computing the derivatives we need. In the following, we omit the dependence on \(x\) when it is clear from the context to simplify the notation.

We now have the following.

Notice that, for the trivial choice \(\rho\equiv 0\), 27 yields 4 . Analogously, \(\varphi \equiv 0\) in 27 yields 3 . To finish this section, we rewrite Theorem 7 in the form of Theorem 1. To do so, we need the following elementary lemma.

We conclude this section with a remark on the equality cases of Theorem 1. As anticipated in Theorem 2, \(\langle \rho, \rho \rangle_{\rm P}=0\) if and only if \(\rho\equiv 0\). Moreover, Livshyts [11] has proved that the equality cases of the Brascamp–Lieb inequality are saturated by constants when the inequality is restricted to compact sets. We shall now show that our new infinitesimal forms 27 and 8 of the Prékopa–Leindler inequality 1 admit further non-trivial equality cases. Observe that, by Lemma 4, equality in 27 implies equality in 8 , but not vice-versa. That is, 8 is sharper than 27 . A full characterization is out of the scope of this paper, and will be the subject of future research.

3 Stability for the Poincaré inequality↩︎

The aim of this section is to prove Theorem 2. We point out that both the stability and the equality cases of 3 could be alternatively obtained by a careful inspection of its proof in [7] via the \(L_2\) method. Here, instead, we propose a different approach based on Theorem 1. Further applications will be shown in the remaining sections.

As a first step, we want to extend the validity of Theorem 1 to encompass two natural Sobolev spaces as the test functions in the inequality 8 . For the convenience of the reader, we quickly recap some basics from the theory of (weighted) Sobolev spaces. Since the admissible measures \(\mu\) in Theorem 1 are absolutely continuous with respect to the Lebesgue measure restricted to \(K\), and all sets involved are compact and sufficiently regular, the standard results from the classical theory of Sobolev spaces extend to our setting with only minor modifications. We refrain from presenting the full theory of traces, limiting the exposition to the essential material needed below. For complete references, see, for instance, [33]–[35].

Let \(K \subset \mathbb{R}^n\) be a smooth convex body and \(u:\mathbb{R}^n \to \mathbb{R}\) a smooth and strictly convex function, and consider on \(K\) the log-concave measure \(\mu\) with density \(\, \mathrm{d}\mu(x)=e^{-u(x)}\, \mathrm{d}x\). For \(\psi \in C^1(K)\), consider the weighted Sobolev norm \[\|\psi\|_{H^{1}(K,\mu)}^{2} = \int_{K} \left(\psi^{2} + \|\nabla\psi\|^{2}\right)\, \, \mathrm{d}\mu,\] where \(\|x \|^2=\langle x, x \rangle\) for every \(x \in \mathbb{R}^n\). The weighted Sobolev space \(H^{1}(K,\mu)\) is defined as the completion of \(C^{\infty}(K)\) under \(\|\cdot\|_{H^{1}(K,\mu)}\). On \(\partial K\) we set, for \(\rho \in C^{1}(\partial K)\), \[\|\rho\|_{H^{1}(\partial K,\mu)}^{2} = \int_{\partial K}\left(\rho^{2} + \|\nabla_{\partial K}\rho\|^{2}\right)\, \, \mathrm{d}\mu.\] Again for \(\rho \in C^{1}(\partial K)\), we denote its fractional Sobolev norm by \[\label{eq:Fractional32Sobolev32norm} \|\rho\|^2_{H^{1/2}(\partial K,\mu)} = \int_{\partial K}\rho^2 \, \mathrm{d}\mu+\int_{\partial K}\!\int_{\partial K} \frac{|\rho(x)-\rho(y)|^{2}}{|x-y|^{n}} \, \mathrm{d}\mu(x) \, \mathrm{d}\mu(y).\tag{32}\] The spaces \(H^{1}(\partial K,\mu)\) and \(H^{1/2}(\partial K,\mu)\) correspond to the completions of the space \(C^{\infty}(\partial K)\) with the respective norms.

For a function \(\psi \in C^{1}(K)\cap H^{1}(K,\mu)\), we can consider the trace operator, which is defined on \(C^{1}(K)\) by \[Tr(\psi) = \psi |_{\partial K}, \,\,\, \forall \psi \in C^{1}(K),\] which extends to \(H^{1}(K,\mu)\) by continuity. Given now a function \(\rho \in H^{1/2}(\partial K,\mu)\), classical results in Sobolev extension theory, see, e.g., [35], guarantee the existence of a constant \(C_{1}=C_{1}(K,\mu)\) and a function \(\tilde{\psi} \in H^{1}(K,\mu)\) such that \(Tr\left(\tilde{\psi}\right) = \rho\) and \[\label{eq:fractional95extension} \|\tilde{\psi}\|_{H^{1}(K,\mu)} \leq C_{1} \,\|\rho\|_{H^{1/2}(\partial K,\mu)}.\tag{33}\] Recall also that the space \(H^{1}(\partial K,\mu)\), under our assumptions on \(\mu\) and \(K\), can be continuously embedded into \(H^{1/2}(\partial K,\mu)\), see e.g. [33]. As a consequence, we may find a constant \(C_{2} = C_{2}(K,\mu)\) such that \[\label{eq:frac95boundary95embedding} \|\rho\|_{H^{1/2}(\partial K,\mu)} \le C_2\,\|\rho\|_{H^{1}(\partial K,\mu)}\tag{34}\] for all \(\rho \in C^1(\partial K)\). For the rest of the section, \(c,\tilde{c},C,\text{etc}\) will denote constants which depend only on the measure \(\mu\) and the convex set \(K\) and whose value may change at different steps.

Since we are interested in applying 8 to functions in \(H^{1}(\partial K,\mu)\) and \(H^{1}(K,\mu)\), let us justify that this passage is indeed possible. Indeed, usual approximation arguments via smooth functions will suffice.

We now obtain a technical estimate, resembling an interpolation inequality.

We can now prove the main result of this section (Theorem 2 in the introduction).

As an immediate corollary, we deduce the equality cases of the weighted Poincaré inequalities in the boundary of convex sets.

The conclusion of Corollary 1 is in stark contrast to the equality cases of the unweighted version established by Colesanti in [6]. The equality cases (or absence thereof) of the weighted Poincaré inequality are evidence of the fact that this inequality can be improved in various ways, as already discussed in [7]. The improvement of these infinitesimal forms is extremely natural (see Proposition 11 later), and lies at the core of the recent treatment of concavity properties of log-concave measures, which will be the topic of the next section.

4 Concavity powers of strictly log-concave measures↩︎

The purpose of this section is to present applications of the inequality in Theorem 1 to concavity properties of functionals restricted to sets in the class \(\mathcal{K}^n\). In the present section it will be convenient to normalize \(\mu\) as a probability measure supported on \(\mathbb{R}^n\). We adopt some terminology and observations from various authors; see, for example, [11] and the references therein. We begin the exposition in a general, not necessarily symmetric, setting.

We are interested in the largest value \(p(\mu)\) such that \[\label{eq:pmu} \mu((1-t)K+tL)^{p(\mu)} \geq (1-t)\mu(K)^{p(\mu)}+t\mu(L)^{p(\mu)}\tag{49}\] for every \(K,L \in \mathcal{K}^n\) and \(t \in [0,1]\). Notice that, as \(\mu\) is log-concave, 49 is, by definition, true for the power \(p=0\). Moreover, Hölder’s inequality implies that if 49 is true for some \(p(\mu)>0\), then it remains true for every \(p \in [0,p(\mu))\). As mentioned in the introduction, it is conjectured that the concavity exponent in 49 is \(1/n\) if one restricts the inequality to an even log-concave probability measure and origin-symmetric convex bodies. That is, 10 holds. This conjecture was verified in the contexts presented in [10], [19], [20], but a full solution is yet to be found.

We briefly explain how estimating \(p(\mu)\) can be approached via a local to global principle, following [11], [22]. Fix a convex set \(K\) and define the concavity power of \(K\) with respect to \(\mu\) by \[\label{eq:local95concavity95power} p(\mu,K)= \limsup_{\varepsilon \rightarrow 0^{+}} \left\{p>0 : \forall L \in \mathcal{K}^{n},\;\mu((1-\varepsilon)K+\varepsilon L)^{p} \geq (1-\varepsilon)\mu(K)^{p}+\varepsilon\mu(L)^{p}\right\}.\tag{50}\] Then it follows that (see, e.g., [11]) \[p(\mu) = \inf_{K \in \mathcal{K}^n} p(\mu,K).\] The symmetric analogues are defined similarly, with the obvious modifications. Therefore, the general strategy is to prove that for every convex set \(K\) one has \(p(\mu,K) \geq c(\mu)\) for some \(c(\mu)\) independent of \(K\), and then to deduce the global concavity estimate 49 .

We now explain how the quantity in 50 is connected to spectral inequalities. Fix a smooth and strictly convex \(K\in\mathcal{K}^n\), and let \(L\in\mathcal{K}^n\) be smooth and strictly convex as well. Define the function \(f : \mathbb{S}^{n-1} \to \mathbb{R}\) by \[\label{eq:addmisible95pertubations} f = h_{L}-h_{K},\tag{51}\] where \(\nu_K\) denotes the outer unit normal to \(\partial K\). Then \[(1-t)K + t L = [h_{K}+tf].\] With this notation, 50 can be rewritten as \[p(\mu,K)= \sup_{p \in (0,\infty)} \left\{ p : \left.\frac{d^{2}}{dt^{2}}\mu(K(f,t))^{p}\right|_{t=0} \leq 0\quad \forall f \in C^{1}(\mathbb{S}^{n-1}), \text{ where } K(f,t)=[h_K+tf] \right\},\] where we have that linear combinations of differences of support functions in \(C^2(\mathbb{S}^{n-1})\) are dense in \(C^1(\mathbb{S}^{n-1})\). With the notation \(\rho=f(\nu_K)\) (which is in \(C ^1(\partial K)\) by the regularity of \(K\)), if we now write \[J(\rho) = \frac{\mu(K)\langle \rho,\rho\rangle_{\rm P}}{\left(\int_{\partial K}\rho\,\, \mathrm{d}\mu\right)^2},\] where we recall that the bilinear form appearing on the right-hand side is the one defined in 5 , explicit computations (compare again [11]) show that \[\label{eq:def95concavity95power} p(\mu,K)= \inf_{\rho \in C^{1}(\partial K)} J(\rho).\tag{52}\]

In the case where the convex sets \(K\) and \(L\) are origin symmetric, the admissible perturbations \(f\) in 51 must be even. Consequently, the infimum in 52 is taken over all even functions in \(C^{1}(\partial K)\). Using the \(L_2\) method, the functional \(J(\rho)\) can be estimated from below using only integrals over \(K\); see, for example, Proposition 3.16 in [11]. This has been the prevailing approach in all prior results in the literature.

As mentioned in the introduction, we shall propose a different approach, working directly with the functional \(J(\rho)\). In particular, we shall prove that the infimum in 52 is in fact attained, and compute the associated Euler–Lagrange equation. By a standard density argument, \[\label{eq:concavity95power95Sobolev} p(\mu,K)= \inf_{\rho \in H^{1}(\partial K,\mu)} J(\rho),\tag{53}\] so that we may work in the Sobolev space \(H^{1}(\partial K,\mu)\), which is better suited to our purposes since it is complete. The idea of deducing concavity principles via an associated partial differential equation is not new; see, for example, [36], [37] for analogous approaches concerning the first Dirichlet eigenvalue of the Laplacian and torsional rigidity, as well as [38] for a broader discussion. However, to the best of our knowledge, this approach is new in the context of concavity powers of convex sets, partly because the corresponding Euler–Lagrange equation had not previously been available. In this sense, our method yields an explicit characterization of the concavity power of a given set \(K\) through a precise perturbation, incorporating the structure of the underlying measure \(\mu\).

4.0.0.1 Existence of a minimizer and the Lax–Milgram theorem.

Now that we have concluded the necessary preparations, we shall prove the existence of a minimizer in 53 . At the core of our approach lies the Lax–Milgram theorem. In order to apply it, we need to show that the symmetric bilinear form \(\langle \cdot, \cdot \rangle_{\rm P}\) is continuous and coercive with respect to the norm \(\|\cdot\|_{H^{1}(\partial K,\mu)}\). Since continuity was established in Lemma 5, it remains to prove coercivity. A fundamental step is the following strengthened form of the Poincaré inequality 3 .

We can now show that \(\langle \cdot, \cdot\rangle_{\rm P}\) is coercive.

The main result of this section (Theorem 3 in the introduction) reads as follows. We recall that for a vector field \(X \in C^1(\partial K,\mathbb{R}^n)\), \(\nabla_{\partial K} \cdot X\) is the tangential divergence of \(X\).

If we assume symmetry with respect to the origin for \(K\) and that \(\mu\) is even, we obtain the following corollary of Theorem 12. These assumptions are precisely those of the dimensional Brunn–Minkowski conjecture 10 , which is known to be false for non-symmetric sets and non-even measures. Compare, e.g., [39].

Observe that, under the assumptions of Corollary 2, we can write 53 as \[p(\mu,K)=\inf \{ J(\rho): \rho \in H^{1}(\partial K,\mu) \text{ and }\rho \text{ is even} \}.\] In the notation of [11], \(p_s(\mu,K)=p(\mu,K)\) or, in other words, restricting the space of admissible perturbations to even ones in 53 is a consequence of the underlying symmetry.

4.0.0.2 A reformulation of the dimensional Brunn–Minkowski conjecture.

We now rewrite 58 . We use the notation \[\nabla_{\partial K,\mu}\cdot X = \nabla_{\partial K} \cdot X-\langle \nabla_{\partial K}u,X\rangle\] for the weighted tangential divergence of a vector field \(X \in C^1(\mathbb{R}^n,\mathbb{R}^n)\). Notice that, by the divergence theorem on a manifold without boundary, the weighted divergence satisfies \[\label{eq:weighted95div} \int_{\partial K} \nabla_{\partial K, \mu} \cdot X \, \mathrm{d}\mu = 0\tag{62}\] for every \(X \in C^1(\mathbb{R}^n,\mathbb{R}^n)\). With this notation, we define on \(H^1(\partial K, \mu)\) the operator \(\mathcal{L}\) in Theorem 12 by \[\label{def:Boundary95Elliptic95Operator} \mathcal{L}(\rho) = -\nabla_{\partial K,\mu}\cdot({\rm II}^{-1}\nabla_{\partial K}\rho) -{\rm H}_\mu \rho +\frac{1}{\mu(K)}\int_{\partial K} \rho\, \mathrm{d}\mu.\tag{63}\] Thus, the function \(\overline{\rho}\) is a weak solution of \(\mathcal{L}(\overline{\rho})=1\). Notice that, if \(K\) is smooth, then the operator \(\mathcal{L}\) is uniformly elliptic since \({\rm II}^{-1}\) is positive definite. Moreover, for every \(\rho \in H^1(\partial K, \mu)\) a quick computation shows that \[\label{eq:P95L95identity} \int_{\partial K} \rho \mathcal{L}(\rho) \, \mathrm{d}\mu = \langle \rho, \rho \rangle_{\rm P}.\tag{64}\]

For a fixed even measure \(\mu\) with smooth and strictly convex potential, 10 is equivalent to proving that, for every smooth origin-symmetric convex body \(K \in \mathcal{K}^n\), \[p(\mu,K) \geq \frac{1}{n}.\] We now provide a reformulation of this statement in which, surprisingly, the bilinear form 7 from 8 appears. As we shall see in the proof, the interaction between \(\mathcal{L}\) and the support function \(h_K\) plays a central role.

We conclude this section with a further remark on the role of \(\mathcal{L}\) and \(h_K\). Consider indeed 67 in the case \(u\equiv 0\). Notice that the operator \(\mathcal{L}\) is still well-defined. Then, immediate computations show that \[\mathcal{L}(h_K)=1.\] This is equivalent to the fact that in Colesanti’s version of 3 (that is, when \(\mu\) is the Lebesgue measure), the equality case is given by homotheties of \(K\). Moreover, denoting by \(\mathrm{vol}_n\) the standard Lebesgue measure on \(\mathbb{R}^n\), it is well known that \[\int_{\partial K} h_K(\nu_K)\, \mathrm{d}\mathcal{H}^{n-1}=n\mathrm{vol}_n(K),\] which encodes the dimensional concavity of convex sets, given by the Brunn–Minkowski inequality. In this sense, for a sufficiently smooth log-concave measure \(\mu\), the corresponding function \(\overline{\rho}\) may be viewed as a weighted analogue of the support function of \(K\).

5 Symmetric case and Hessian pinching↩︎

In this final section, we shall prove Theorem 4. To do so, we specialize to the case where the convex body \(K\) is origin-symmetric and the measure \(\mu\) is even. First we report the following lemma, which is a direct consequence of [22].

Combining Theorem 7 with 69 , we obtain the following general bound.

We remark that the above estimate cannot lead to a resolution of the dimensional Brunn–Minkowski conjecture, since even applying the main inequality from Theorem 1 already yields a bound strictly larger than \(n\). Nevertheless, the estimate holds for an arbitrary convex body \(K\), which need not contain the origin, and for an arbitrary smooth log-concave measure \(\mu=e^{-u}\). In this general setting, we obtain control of the concavity power in terms of the quantity \[\frac{1}{\mu(K)}\int_{K}(\nabla^{2}u)^{-1}\nabla u,\nabla u\rangle\, \mathrm{d}\mu.\] Assuming, in addition, that \(K\) is symmetric and that the Hessian of \(u\) is pinched, we can finally prove Theorem 4. We repeat the statement for the convenience of the reader.

5.0.0.1 Acknowledgments.

The first-named author was supported by the Austrian Science Fund (FWF): 10.55776/P36344N. The second-named author was supported, in part, by the Austrian Science Fund (FWF): 10.55776/PAT3787224.

Sotiris Armeniakos
Institut für Stochastik und Wirtschaftsmathematik
TU Wien
Wiedner Hauptstraße 8-10/1046
1040 Wien, Austria
e-mail: sotirios.armeniakos@tuwien.ac.at

Jacopo Ulivelli
Institut für Diskrete Mathematik und Geometrie
TU Wien
Wiedner Hauptstraße 8-10/1046
1040 Wien, Austria
e-mail: jacopo.ulivelli@tuwien.ac.at

References↩︎