Nikos Dafnis, Grigoris Paouris
We prove a sharp moment inequality for a log-concave or a log-convex function, on Gaussian random vectors. As an application we take a stability result for the classical logarithmic Sobolev inequality of L. Gross in the case where the function is log-concave.
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We prove that the exponent of the entropy of one dimensional projections of a log-concave random vector defines a 1/5-seminorm. We make two conjectures concerning reverse entropy power inequalities in the log-concave setting and discuss some examples.
Stability version of the Prekopa-Leindler inequality for log-concave functions on the n-dimensional Euclidean space is established.
We develop in this paper an amelioration of the method given by S. Bobkov and M. Ledoux in GAFA (2000). We prove by Prekopa-Leindler Theorem an optimal modified logarithmic Sobolev inequality adapted for all log-concave measure on $\dR^n$. This inequality implies results proved by Bobkov and Ledoux, the Euclidean Logarithmic Sobolev inequality generalized in the last years and it also implies some convex logarithmic Sobolev inequalities for large entropy.
We prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log-concave measures (that we introduce). These inequalities imply (up to universal constants) the log-Sobolev inequalities proved by Bobkov--Ledoux as well as the isoperimetric inequalities due to Bakry-Ledoux and Bobkov--Zegarlinski. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov--Milman.
The Brunn-Minkowski and Prékopa-Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions so it seems that convexity has no special signficance. On the other hand, it was recently shown that the Brunn-Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. We show that for the subclass of $\log$-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prékopa-L…
We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel sets in $\mathbb{R}^n$ by a $s$-concave probability. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a concentration inequality of Guédon. Applying our inequality to the level sets of functions satisfying a Remez type inequality, we deduce, as it is classical, that these functions enjoy dimension free distribution inequalities and Kahane-Khintchine type inequalities with positive and negative exponent, with respect to an arbitrary $s$-concave probability.
We prove an isoperimetric inequality for probability measures $μ$ on $\mathbb{R}^n$ with density proportional to $\exp(-ϕ(λ| x|))$, where $|x|$ is the euclidean norm on $\mathbb{R}^n$ and $ϕ$ is a non-decreasing convex function. It applies in particular when $ϕ(x)=x^α$ with $α\ge1$. Under mild assumptions on $ϕ$, the inequality is dimension-free if $λ$ is chosen such that the covariance of $μ$ is the identity.
We investigate the role of convexity in Rényi entropy power inequalities. After proving that a general Rényi entropy power inequality in the style of Bobkov-Chistyakov (2015) fails when the Rényi parameter $r\in(0,1)$, we show that random vectors with $s$-concave densities do satisfy such a Rényi entropy power inequality. Along the way, we establish the convergence in the Central Limit Theorem for Rényi entropies of order $r\in(0,1)$ for log-concave densities and for compactly supported, spherically symmetric and unimodal densities, complementing a celebrated result of Barron (1986). Additiona…
An example of a non-log-concave distribution where the difference has a log-concave density
本文通过具体例子证明，非对数凹分布的差值不一定具有对数凹密度，推翻了Prékopa-Leindler不等式的逆向结论。
Geometric and Functional Inequalities for Log-concave Probability Sequences
该研究针对对数凹概率序列，证明了整数集上的伸缩不等式及对应的泛函偏差不等式，揭示对数仿射序列为相关集合极点，借此简化已有卷积不等式证明并得到离散Prékopa-Leindler不等式。
Majorization, Entropy and Concentration Inequalities for Discrete $α$-Concave Random Variables
针对离散 $\alpha$-凹随机变量（$-1/3<\alpha<0$），提出最大方差不等式及凸优超原理，推广对数凹情形，并导出逆向熵幂不等式、Lévy 集中函数界及多项式型集中不等式。
We establish quantitative stability results for the entropy power inequality (EPI). Specifically, we show that if uniformly log-concave densities nearly saturate the EPI, then they must be close to Gaussian densities in the quadratic Wasserstein distance. Further, if one of the densities is log-concave and the other is Gaussian, then the deficit in the EPI can be controlled in terms of the $L^1$-Wasserstein distance. As a counterpoint, an example shows that the EPI can be unstable with respect to the quadratic Wasserstein distance when densities are uniformly log-concave on sets of measure arb…
We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration bounds. As an application, we derive concentration bounds for the intrinsic volumes of a convex body, which generalizes and improves a result of Lotz, McCoy, Nourdin, Peccati, and Tropp (2019).
Two-sided bounds are explored for concentration functions and Rényi entropies in the class of discrete log-concave probability distributions. They are used to derive certain variants of the entropy power inequalities.
A Note on Statistical Distances for Discrete Log-Concave Measures
研究离散对数凹分布上总变差、Wasserstein与$f$-散度等标准统计距离的等价性。
An invariant variational proof for canonical processes with independent symmetric log-concave tails
本文通过不变变分方法，证明了无需加倍假设的独立对称对数凹尾典范过程的指定指数律比较。
We study how well moments of sums of independent symmetric random variables with logarithmically concave tails may be approximated by moments of Gaussian random variables.
We derive two-sided bounds for moments of linear combinations of coordinates od unconditional log-concave vectors. We also investigate how well moments of such combinations may be approximated by moments of Gaussian random variables.
One-sided concentration near the mean of log-concave distributions
本文研究了对数凹分布均值附近的一侧浓度特性，得出了在EX=0和EX²=1条件下概率P(0<X<δ)的下界。
We show that the conjecture of Kannan, Lovász, and Simonovits on isoperimetric properties of convex bodies and log-concave measures, is true for log-concave measures of the form $ρ(|x|_B)dx$ on $\mathbb{R}^n$ and $ρ(t,|x|_B) dx$ on $\mathbb{R}^{1+n}$, where $|x|_B$ is the norm associated to any convex body $B$ already satisfying the conjecture. In particular, the conjecture holds for convex bodies of revolution.
We show that a measure on the real line that is the image of a log-concave measure under a polynomial of degree $d$ possesses a density from the Nikol'skii--Besov class of fractional order $1/d$. This result is used to prove an estimate of the total variation distance between such measures in terms of the Fortet--Mourier distance.
We show that for every mean zero log-concave real random variable $X$ one has $\|X\|_p \leq \frac{p}{q} \|X\|_q$ for $p \geq q \geq 1$, going beyond the well-known case of symmetric random variables. We also prove that in the class of arbitrary log-concave real random variables for $p>q > 0$ the quantity $\|X\|_p / \|X\|_q$ is maximized for some shifted exponential distribution. Building upon this we derive the bound $\|X\|_p \leq C_0 \frac{p}{q} \|X\|_q$ for arbitrary log-concave $X$, with best possible absolute constant $C_0=e^{W(1/e)} \approx 1.3211$ in front of $\frac{p}{q}$, where $W$ sta…
The Paouris inequality gives the large deviation estimate for Euclidean norms of log-concave vectors. We present a modified version of it and show how the new inequality may be applied to derive tail estimates of l_r-norms and suprema of norms of coordinate projections of isotropic log-concave vectors.
We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of expectation of the supremum of "symmetric exponential" processes compared to the Gaussian ones in the Chevet inequality. This is used to give sharp upper estimate for a quantity $Γ_{k,m}$ that controls uniformly the Euclidean operator norm of the sub-matrices with $k$ rows and $m$ columns of an isotropic log-concave unconditional random matrix. We apply these estimates to give a sharp bound for the Restricted Isometry Constant of a random matrix with ind…
A reverse entropy power inequality for i.i.d. log-concave random variables
研究独立同分布对数凹随机变量的反向熵幂不等式，通过递减重排、控制和测度变换等工具，证明\(h_\infty(X + Y) \leq h_\infty(Z + W)\)，并得到整数值单调对数凹随机变量类似结果。
A note on the Littlewood-Offord problem for discrete log-concave distributions
本文扩展了Littlewood-Offord问题，研究离散对数凹分布下的熵功率不等式，并探讨了算术级数和熵版本的问题。
Extending results of Harg{é} and Hu for the Gaussian measure, we prove inequalities for the covariance Cov$_μ(f, g)$ where $μ$ is a general product probability measure on $\mathbb{R}^d$ and $f,g: \mathbb{R}^d \to \mathbb{R}$ satisfy some convexity or log-concavity assumptions, with possibly some symmetries.
We show that every symmetric random variable with log-concave tails satisfies the convex infimum convolution inequality with an optimal cost function (up to scaling). As a result, we obtain nearly optimal comparison of weak and strong moments for symmetric random vectors with independent coordinates with log-concave tails.
We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by independent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.
Sharp Rosenthal-type inequalities for mixtures and log-concave variables
本文为固定分布混合的独立随机变量和矩建立了带尖锐常数的 Rosenthal 型不等式，并在对数凹情形下确定了极值元。
$L^\infty$-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model
该研究证明各向异性强对数凹密度最优输运的$L^\infty$-Wasserstein距离上界，改进输运-信息不等式，并用Langevin动力学方法将Fisher无穷小模型的指数收敛结果从一维推广到任意维。
A note on the maximum probability of ultra log-concave distributions
本文指出，超对数凹分布的最大概率不小于同均值泊松分布最大概率的结论，在均值大于1时并非普遍成立。
In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \[ μ(λA + (1-λ)B)^{1/n} \geq λμ(A)^{1/n} + (1-λ)μ(B)^{1/n} \] holds true for an unconditional product measure $μ$ with decreasing density and a pair of unconditional convex bodies $A,B \subset \mathbb{R}^n$. We also show that the above inequality is true for any unconditional $\log$-concave measure $μ$ and unconditional convex bodies $A,B \subset \mathbb{R}^n$. Finally, we prove that the inequality is true for a symmetric $\log$-concave measu…
The first step to study lower bounds for a stochastic process is to prove a special property - Sudakov minoration. The property means that if a certain number of points from the index set are well separated then we can provide an optimal type lower bound for the mean value of the supremum of the process. Together with the generic chaining argument the property can be used to fully characterize the mean value of the supremum of the stochastic process. In this article we prove the property for canonical processes based on radial-type log concave measures.
We review some simple techniques based on monotone mass transport that allow to obtain transport-type inequalities for any log-concave probability measure, and for more general measures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp-Lieb variance inequality.
Entropy-variance inequalities for discrete log-concave random variables via degree of freedom
本文利用离散自由度概念证明整数值对数凹随机变量的精确最小熵-方差不等式，证明几何分布能最小化最小熵，并由此推导出改进的离散 Rényi 熵幂不等式。
We prove stability estimates for the Shannon-Stam inequality (also known as the entropy-power inequality) for log-concave random vectors in terms of entropy and transportation distance. In particular, we give the first stability estimate for general log-concave random vectors in the following form: for log-concave random vectors $X,Y \in \mathbb{R}^d$, the deficit in the Shannon-Stam inequality is bounded from below by the expression $$ C \left(\mathrm{D}\left(X||G\right) + \mathrm{D}\left(Y||G\right)\right), $$ where $\mathrm{D}\left( \cdot ~ ||G\right)$ denotes the relative entropy with resp…
In the paper we pursue the analysis from the section 5 of the Talagrand's paper "Sample boundedness of stochastic processes under increment conditions." Ann. Probab. 18, No. 1, 1-49. In particular we give the proof of some Sobolev Inequality and then apply it to obtain if and only if condition for all processes with bounded icrements to have bounded samples. The processes are defined on a compact, concave subspaces of $\R^n$ with a metric $d(s,t)=η(||s-t||)$, where $η$ is concave and $||.||$ is a norm on $\R^n$.
A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures
本文基于维度Brunn-Minkowski不等式，为log-concave测度建立了通用界，改进了相关猜想，并在特殊测度类别中取得进展。
We prove, using optimal transport tools, weighted Poincar'e inequalities for log-concave random vectors satisfying some centering conditions. We recover by this way similar results by Klartag and Barthe-Cordero-Erausquin for log-concave random vectors with symmetries. In addition, we prove that the variance conjecture is true for increments of log-concave martingales.
We discuss interplays between log-concave functions and log-concave sequences. We prove a Bernstein-type theorem, which characterizes the Laplace transform of log-concave measures on the half-line in terms of log-concavity of the alternating Taylor coefficients. We establish concavity inequalities for sequences inspired by the Prékopa-Leindler and the Walkup theorems. One of our main tools is a stochastic variational formula for the Poisson average.
On measures strongly log-concave on a subspace
研究仅在方向子空间上弯曲的对数凹测度的集中性质，采用适配的随机局部化过程进行证明。
The paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the $L^2$-distance between these polynomials.
We establish upper bounds for tails of order statistics of isotropic log-concave vectors and apply them to derive a concentration of l_r norms of such vectors.
We study the log-concave measures, their characterization via the Prékopa-Leindler property and also define a subset of it whose elements are called super log-concave measures which have the property of satisfying a logarithmic Sobolev inequality. We give some results about their stability. Certain relations with measure transportation of Monge-Kantorovitch and the Monge-Ampére equation are also indicated with applications.
The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincar{é} constant. First we revisit E. Milman's result [40] on the link between weak (Poincar{é} or concentration) inequalities and Cheeger's inequality in the logconcave cases, in particular extending localization ideas and a result of Latala, as well as providing a simpler proof of the nice Poincar{é} (dimensional) bound in the inconditional case. Then we prove alternative transference principle by concentration or using various distances (total variation, Wasserstein…
We formulate and discuss a conjecture concerning lower bounds for norms of log-concave vectors, which generalizes the classical Sudakov minoration principle for Gaussian vectors. We show that the conjecture holds for some special classes of log-concave measures and some weaker forms of it are satisfied in the general case. We also present some applications based on chaining techniques.
We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $|X|$ of an isotropic log-concave random vector $X\in\R^n$, stating that for every $t\geq 1$, $P(|X|\geq ct\sqrt n)\leq \exp(-t\sqrt n)$. More precisely we show that for any log-concave random vector $X$ and any $p\geq 1$, $(E|X|^p)^{1/p}\sim E |X|+\sup_{z\in S^{n-1}}(E |< z,X>|^p)^{1/p}$.
Stability of Klartag's improved Lichnerowicz inequality
研究Klartag改进的Lichnerowicz不等式，分析等式情形并证明无高斯因子时界可进一步改进，且对有限Fisher信息测度给出定量改进。
We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.
High-Dimensional Ultra-Log-Concave Distributions
本文提出高维超对数凹分布的定量推广（$\delta$-超对数凹性），建立函数不等式、集中性与最大熵原理，并应用于多个模型。
A concentration property of the functional ${-}\log f(X)$ is demonstrated, when a random vector X has a log-concave density f on $\mathbb{R}^n$. This concentration property implies in particular an extension of the Shannon-McMillan-Breiman strong ergodic theorem to the class of discrete-time stochastic processes with log-concave marginals.
We develop an information-theoretic perspective on some questions in convex geometry, providing for instance a new equipartition property for log-concave probability measures, some Gaussian comparison results for log-concave measures, an entropic formulation of the hyperplane conjecture, and a new reverse entropy power inequality for log-concave measures analogous to V. Milman's reverse Brunn-Minkowski inequality.
A Chaining Approach to Canonical Processes with Log-Concave Tails
本文证明了无需Δ2条件的对数凹尾典型过程的分解定理，采用基于可容许划分和切割映射的经典链式方法，并恢复了控制测度刻画。
We show the existence of the strong solutions of the Monge-Ampere equation for the log-concave measures without any regularity assumption. In particular the measures with density of the form $\exp -f$, where $f$ is an $H$-convex function which may take the value $\infty$ on a set of positive measure are included.
Minimum entropy of a log-concave variable for fixed variance
本文证明固定方差下对数凹变量的最小熵由指数分布达到，并据此推导加性噪声信道容量上界及改进反向熵幂不等式常数。
Bi-log-concavity of probability measures is a univariate extension of the notion of log-concavity that has been recently proposed in a statistical literature. Among other things, it has the nice property from a modelisation perspective to admit some multimodal distributions, while preserving some nice features of log-concave measures. We compute the isoperimetric constant for a bi-log-concave measure, extending a property available for log-concave measures. This implies that bi-log-concave measures have exponentially decreasing tails. Then we show that the convolution of a bi-log-concave measu…
A note on the spectral gap for log-concave probability measures on convex bodies
本文研究了凸体上对数凹概率测度的谱隙问题，提出了基于分布参数、维度、域几何特征和原点位置等量的显式下界，并通过经典与非经典例子验证了结果。
We show that the modified log-Sobolev constant for a natural Markov chain which converges to an $r$-homogeneous strongly log-concave distribution is at least $1/r$. Applications include a sharp mixing time bound for the bases-exchange walk for matroids, and a concentration bound for Lipschitz functions over these distributions.
We derive two concentration inequalities for linear functions of log-concave distributions: an enhanced version of the classical Brascamp--Lieb concentration inequality, and an inequality quantifying log-concavity of marginals in a manner suitable for obtaining variance and tail probability bounds. These inequalities are applied to the statistical mechanics problem of estimating the fluctuations of random surfaces of the $\nablaφ$ type. The classical Brascamp--Lieb inequality bounds the fluctuations whenever the interaction potential is uniformly convex. We extend these bounds to the case of c…
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