Karoly J. Boroczky, Pavlos Kalantzopoulos, Dongmeng Xi
The works of Bennett, Carbery, Christ, Tao and of Valdimarsson have clarified when equality holds in the Brascamp-Lieb inequality. Here we characterize the case of equality in the Geometric case of Barthe's reverse Brascamp-Lieb inequality.
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We give a short proof of the Brascamp-Lieb theorem, which asserts that a certain general form of Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell's stochastic proof of the Prékopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.
A new proof is given for the fact that centered gaussian functions saturate the Euclidean forward-reverse Brascamp-Lieb inequalities, extending the Brascamp-Lieb and Barthe theorems. A duality principle for best constants is also developed, which generalizes the fact that the best constants in the Brascamp-Lieb and Barthe inequalities are equal. Finally, as the title hints, the main results concerning finiteness, structure and gaussian-extremizability for the Brascamp-Lieb inequality due to Bennett, Carbery, Christ and Tao are generalized to the setting of the forward-reverse Brascamp-Lieb ine…
From the Sphere to the Grassmannian: Continuous Brascamp--Lieb Inequalities via Measure Approximation and Heat Flow
本文研究球面与格拉斯曼流形上各向同性测度的连续Brascamp--Lieb不等式，通过离散逼近与热半群保持两种方法给出证明，并概述混合秩情形。
Regularized Brascamp--Lieb inequalities
本文采用热流方法研究正则化输入函数下的逆向Brascamp–Lieb不等式，将Barthe–Wolff定理推广到该情形，并在正-逆向Brascamp–Lieb不等式框架下推导出正则化不等式，重新得到相关高斯饱和性质。
In this work we consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities. For this we use optimal transport methods and the Borell-Brascamp-Lieb inequality. These refinements can be written as a deficit in the classical inequalities. They have the right scale with respect to the dimension. They lead to sharpened concentration properties as well as refined contraction bounds, convergence to equilibrium and short time behaviour for the laws of solutions to stochastic differential equations.
This paper builds upon several recent works, where semigroup proofs of Brascamp-Lieb inequalities are provided in various settings (Euclidean space, spheres and symmetric groups). Our aim is twofold. Firstly, we provide a general, unifying, framework based on Markov generators, in order to cover a variety of examples of interest going beyond previous investigations. Secondly, we put forward the combinatorial reasons for which unexpected exponents occur in these inequalities.
$L^2$-stability for the variance Brascamp-Lieb inequality
本文通过额外假设证明了方差Brascamp-Lieb不等式的$L^2$稳定性估计，基于Machado和Ramos的$L^1$稳定性定理。
We prove a general duality result showing that a Brascamp--Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This open a new approach to the proof of Brascamp--Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on $\R^n$, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of th…
We explore the consequences of the so-called intertwinings between gradients and Markov diffusion operators on $R^d$ in terms of second-order Brascamp-Lieb inequalities for log-concave distributions and beyond, extending our inequalities established in a previous paper. As a result, we derive some convenient lower bounds on the $(d+1)^{th}$ positive eigenvalue depending on the spectral gap of the dual Markov diffusion operator given by the intertwining. To see the relevance of our approach, we apply our spectral results in the case of perturbed product measures, freeing us from Helffer's class…
Gaussian Correlation via Inverse Brascamp-Lieb
本研究借助推广的Nakamura-Tsuji对称逆Brascamp-Lieb不等式，给出Royen高斯相关性不等式的简洁替代证明，明确该逆不等式是正向不等式的对偶对应，且对数凹性假设不可省略。
We prove a reverse form of the multidimensional Brascamp-Lieb inequality. Our method also gives a new way to derive the Brascamp-Lieb inequality and is rather convenient for the study of equality cases.
The goal of the present paper is to discuss new transport inequalities for convex measures. We retrieve some dimensional forms of Brascamp-Lieb inequalities. We also give some quantitative forms involving the Wasserstein's distances.
We give a new approach, inspired by Hörmander's $L^2$-method, to weighted variance inequalities which extend results obtained by Bobkov and Ledoux. It provides in particular a local proof of the dimensional functional forms of the Brunn-Minkowski inequalities. We also present several applications of these variance inequalities, including reverse Hölder inequalities for convex functions, weighted Brascamp-Lieb inequalities and sharp weighted Poincaré inequalities for generalized Cauchy measures.
Brascamp--Lieb inequalities on locally compact abelian groups
本文在局部紧阿贝尔群上建立Brascamp--Lieb常数的结构定理，统一并推广了欧氏空间与有限生成群情形的有限性刻画，并突出其Fourier不变性。
A logarithmic Brascamp--Lieb variance inequality
本文针对满足局部对数Brunn–Minkowski不等式的偶对数凹测度，证明了强化的对数型Brascamp–Lieb方差不等式，采用次水平体积变分等方法，得到齐次势函数的尖锐径向余项。
In this paper, we derive sharp nonlinear dimension-free Brascamp--Lieb inequalities (including hypercontractivity inequalities) for distributions on Polish spaces, which strengthen the classic Brascamp--Lieb inequalities. Applications include the extension of Mrs. Gerber's lemma to the cases of Rényi divergences and distributions on Polish spaces, the strengthening of small-set expansion theorems, and the characterization of the exponent of the $q$-stability. Our proofs in this paper are based on information-theoretic and coupling techniques.
We propose a new, self-contained, approach to H. Raufi's extension of Prekopa's theorem for matrix-valued log-concave functions. Along the way, new related inequalities are established, in particular a Brascamp-Lieb variance inequality for matrix weights.
On weighted Blaschke--Santalo and strong Brascamp--Lieb inequalities
本文研究函数型Blaschke-Santaló不等式的新推广，探索其在标准高斯测度经典设定外的应用，涉及加权Blaschke–Santalo不等式与强Brascamp–Lieb不等式。
We strengthen, in two different ways, the so called Borell-Brascamp- Lieb inequality in the class of power concave functions with compact support. As examples of applications we obtain two quantitative versions of the Brunn- Minkowski inequality and of the Urysohn inequality for torsional rigidity.
We study stability issues for the so-called Borell-Brascamp-Lieb inequalities, proving that when near equality is realized, the involved functions must be $L^1$-close to be $p$-concave and to coincide up to homotheties of their graphs.
Regularized Brascamp-Lieb inequalities via Optimal Transport and Study of Equality Cases
本文利用最优传输理论研究正则化的Brascamp-Lieb不等式，探讨了其常数有限性和高斯极值存在的问题，并通过热流方法找到所有极值器，同时给出了一些有趣的应用。
Brascamp--Lieb inequalities for fractal dimensions
本文利用泛函分析中的Brascamp--Lieb不等式，证明了关于分形集的上盒维数、包络维数和Assouad维数的新不等式，这些不等式以某些投影的维数为条件。与Hausdorff和下盒维数不同，类似不等式不成立。作者将这些分形Brascamp--Lieb不等式应用于正交投影的例外集估计，并为某些受限和集提供精确的维数估计，同时通过非线性Brascamp--Lieb不等式建立了类似的非线性不等式。
We formulate a non-commutative analog of the Brascamp-Lieb inequality, and prove it in several concrete settings.
Adjoint Brascamp-Lieb inequalities
本文提出 Brascamp-Lieb 不等式的伴随形式，将其刻画为熵版本的 $L^p$ 对应物，并用于重证 Gowers 范数对数凸性及建立层析变换的反向不等式。
In this paper, we derive from the supersymmetry of the Witten Laplacian Brascamp-Lieb's type inequalities for general differential forms on compact Riemannian manifolds with boundary. In addition to the supersymmetry, our results essentially follow from suitable decompositions of the quadratic forms associated with the Neumann and Dirichlet self-adjoint realizations of the Witten Laplacian. They moreover imply the usual Brascamp-Lieb's inequality and its generalization to compact Riemannian manifolds without boundary.
Quantitative stability for the Brascamp-Lieb inequality and moment measures
本文通过Prékopa-Leindler不等式的稳定性版本，得到Brascamp-Lieb不等式的精确稳定性结果，并探讨了矩测度的稳定性问题，强调了稳定性常数与凸函数无关的创新特性。
We provide a number of new quantitative versions of Helly's theorem. For example, we show that for every family $\{P_i:i\in I\}$ of closed half-spaces $$P_i=\{x\in {\mathbb R}^n:\langle x,w_i\rangle \leq 1\}$$ in ${\mathbb R}^n$ such that $P=\bigcap_{i\in I}P_i$ has positive volume, there exist $s\leq αn$ and $i_1,\ldots, i_s\in I$ such that $$|P_{i_1}\cap\cdots\cap P_{i_s}|\leq (Cn)^n\,|P|,$$ where $α, C>0$ are absolute constants. These results complement and improve previous work of Bárány-Katchalski-Pach and Naszódi. Our method combines the work of Srivastava on approximate John's decomposi…
We initiate the study of a duality theory which applies to norm inequalities for pointwise weighted geometric means of positive operators. The theory finds its expression in terms of certain pointwise factorisation properties of function spaces which are naturally associated to the norm inequality under consideration. We relate our theory to the Maurey-Nikisin-Stein theory of factorisation of operators, and present a fully multilinear version of Maurey's fundamental theorem on factorisation of operators through $L^1$. The development of the theory involves convex optimisation and minimax theor…
An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the $L^2$ norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto \cite{OM} proved a variant of this with the two $L^2$ norms replaced by $L^1$ and $L^\infty$ norms, but only for $\R^1$. We prove a generalization of both by extending these inequalities to $L^p$ and $L^q$…
On p-Brunn-Minkowski and Brascamp-Lieb inequalities
本文证明了对称 p-布朗-闵科夫斯基不等式的新充分条件，并在所有维度中建立了 L_q 球体的局部对数布朗-闵科夫斯基不等式。
From Brunn-Minkowski to Prékopa-Leindler and Borell-Brascamp-Lieb: discrete inequalities
本文通过Brunn-Minkowski不等式推导出Prékopa-Leindler和Borell-Brascamp-Lieb不等式，并证明了其在整数空间上的离散形式。
Sharp Quantitative Stability for the Prékopa-Leindler and Borell-Brascamp-Lieb Inequalities
本文构建统一框架，在完全一般条件下证明 Borell-Brascamp-Lieb 不等式的最优定量稳定性，并据此解决 Prékopa-Leindler 不等式的尖锐稳定性猜想。
A Rogers--Brascamp--Lieb--Luttinger inequality in the space of matrices
本文在矩阵空间 $M_{n,m}(\mathbb R)$ 中，针对纤维对称化证明了拟凹函数的 Rogers--Brascamp--Lieb--Luttinger 型不等式，并给出应用。
On plurisubharmonic Gaussian correlation of Barthe and Cordero-Erausquin
本文利用复 Gaussian 情形中的二阶分部积分公式，给出 Barthe–Cordero-Erausquin 多复次调和函数 Gaussian 相关性不等式的简化证明。
We survey several Talagrand type inequalities and their application to influences with the tool of hypercontractivity for both discrete and continuous, and product and non-product models. The approach covers similarly by a simple interpolation the framework of geometric influences recently developed by N. Keller, E. Mossel and A. Sen. Geometric Brascamp-Lieb decompositions are also considered in this context.
Operator capacity, the Brascamp--Lieb inequality and geometric programming
本文通过几何规划视角探讨算子容量与布拉斯坎普-利布常数，提出新的研究成果，包括算子容量的近似极小值和局部霍尔德正则性，以及推广到有向图数据容量的可能。
It is known that by dualizing the Bochner-Lichnerowicz-Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). When the manifold has a boundary, an appropriate generalization of the Reilly formula may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichl…
Functional Liftings of Restricted Geometric Inequalities
本文提出广义上卷积方法，将受限几何不等式提升为函数不等式，并在 Gaussian、对数情形及幂零 Lie 群中给出应用。
We provide variants and improvements of the Brascamp-Lieb variance inequality which take into account the invariance properties of the underlying measure. This is applied to spectral gap estimates for log-concave measures with many symmetries and to non-interacting conservative spin systems.
Borell--Brascamp--Lieb inequality with finitely many output functions
本文将经典Borell–Brascamp–Lieb不等式扩展为每个输入函数对应不同输出函数的形式，建立了含加权幂平均、幂变换的新不等式，其特例可还原经典不等式或对应多输出Prékopa–Leindler不等式的特例。
We show that the sharp Young's inequality for convolutions first obtained by Bechner and Brascamp-Lieb can be derived from the monotone in time evolution of a Lyapunov functional of the convolution of two solutions to the heat equation, with different diffusion coefficients, first introduced by Bennett and Bez. Our proof is based on a suitable adaptation of an old idea of Stam and Blachman, used to obtain Shannon's entropy power inequality.
In this article, we give a complex-geometric proof of the Alexandrov-Fenchel inequality without using toric compactifications. The idea is to use the Legendre transform and develop the Brascamp-Lieb proof of the Prékopa theorem. New ingredients in our proof include an integration of Timorin's mixed Hodge-Riemann bilinear relation and a mixed norm version of Hörmander's $L^2$-estimate, which also implies a non-compact version of the Khovanskiĭ-Teissier inequality.
We review some topics in the theory of symmetric decreasing rearrangements with a particular focus on Lieb's fundamental contributions. Topics covered include the Brascamp--Lieb--Luttinger theorem, the sharp Young and Hardy--Littlewood--Sobolev inequalities, as well as the continuity of the rearrangement map on Sobolev spaces.
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