arXiv subjects
Publications and source records attributed to Cyril Roberto.
We prove an $L^2$-stability estimate for the variance Brascamp-Lieb inequality [J. Funct. Anal. 22 (4), 366-389 (1976)] by bootstrapping the recent $L^1$-stability theorem of Machado and Ramos [arXiv:2511.22636] under an additional assumption, which we call the super-Brascamp-Lieb inequality, of independent interest.
The entropy power inequality for independent random vectors is a foundational result of information theory, with deep connections to probability and geometric functional analysis. Several extensions of the entropy power inequality have been developed for settings with dependence, including by Takano, Johnson, and Rioul. We extend these works by developing a quantitative version of the entropy power inequality for dependent random vectors. A notable consequence is that an entropy power inequality stated using conditional entropies holds for random vectors whose joint density is log-supermodular.
We study the behavior of log-supermodular functions under convolution. In particular we show that log-concave product densities preserve log-supermodularity, confirming in the special case of the standard Gaussian density, a conjecture of Zartash and Robeva. Additionally, this stability gives a ``conditional'' entropy power inequality for log-supermodular random variables. We also compare the Ahlswede-Daykin four function theorem and a recent four function version of the Prekopa-Leindler inequality due to Cordero-Erausquin and Maurey and giving transport proofs for the two theorems. In the Prekopa-Leindler case, the proof gives a generalization that seems to be new, which interpolates the classical three and the recent four function versions.
We investigate what we term "generalized sup-convolutions". We show that functional inequalities that enjoy an interpretation as sup-convolution inequalities can be deduced from the special case of indicator functions corresponding to a geometric inequality. As consequences we derive a Borell-Brascamp Lieb inequality for the Gaussian Brunn-Minkowski inequality and give a functional analog of the log-Brunn Minkowski conjecture. Though we focus on Euclidean applications, our results are general and can be directly applied in more abstract settings, like groups or even topological measure spaces without algebraic structure, we instantiate this claim with a Borell-Brascamp-Lieb type inequality for nilpotent Lie groups.
We develop a general comparison result for inf convolution operators related to rearrangement. As a consequence we derive comparison results under spherically symmetric rearrangement for Laplace and polar transforms. As a by product we simplify existing proofs related to the functional Blaschke Santalo inequality of Keith Ball and derive a comparison result for some parabolic PDE.
Inspired by the approach of Ivanisvili and Volberg towards functional inequalities for probability measures with strictly convex potentials, we investigate the relationship between curvature bounds in the sense of Bakry-Emery and local functional inequalities. We will show that not only is the earlier approach for strictly convex potentials extendable to Markov semigroups and simplified through use of the $\Gamma$-calculus, providing a consolidating machinery for obtaining functional inequalities new and old in this general setting, but that a converse also holds. Local inequalities obtained are proven equivalent to Bakry-Emery curvature. Moreover we will develop this technique for metric measure spaces satisfying the RCD condition, providing a unified approach to functional and isoperimetric inequalities in non-smooth spaces with a synthetic Ricci curvature bound. Finally, we are interested in commutation properties for semi-group operators on $\mathbb{R}^n$ in the absence of positive curvature, based on a local eigenvalue criteria.
We show that for log-concave real random variables with fixed variance the Shannon differential entropy is minimized for an exponential random variable. We apply this result to derive upper bounds on capacities of additive noise channels with log-concave noise. We also improve constants in the reverse entropy power inequalities for log-concave random variables.
We revisit several results on exponential integrability in probability spaces and derive some new ones. In particular, we give a quantitative form of recent results by Cianchi-Musil and Pick in the framework of Moser-Trudinger-type inequalities, and recover Ivanisvili-Russell's inequality for the Gaussian measure. One key ingredient is the use of a dual argument, which is new in this context, that we also implement in the discrete setting of the Poisson measure on integers.
It is well known that some important Markov semi-groups have a "regularization effect" -- as for example the hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in $L^p$ for $p>1$. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property for functions that are just integrable, but this conjecture remains open. Nonetheless, the Gaussian analogue of this conjecture was proven in recent years by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of the Ornstein-Uhlenbeck semi-group with a new deviation inequality for log-semi-convex functions under Gaussian measure. In this work, we explore the question of how much more general this phenomenon is. Specifically, our first goal is to explore the validity of both these ingredients for some diffusion semi-groups in $\mathbb{R}^n$, as well as for the $M/M/\infty$ queue on the non-negative integers and the Laguerre semi-groups on the positive real line. Our second goal is to prove a one-dimensional regularization effect for these settings, even in those cases where these ingredients are not valid.
We introduce a transport-majorization argument that establishes a majorization in the convex order between two densities, based on control of the gradient of a transportation map between them. As applications, we give elementary derivations of some delicate Fourier analytic inequalities, which in turn yield geometric "slicing-inequalities" in both continuous and discrete settings. As a further consequence of our investigation we prove that any strongly log-concave probability density majorizes the Gaussian density and thus the Gaussian density maximizes the Rényi and Tsallis entropies of all orders among all strongly log-concave densities.
We prove a quantitative form of the celebrated Ball's theorem on cube slicing in $\mathbb{R}^n$ and obtain, as a consequence, equality cases in the min-entropy power inequality. Independently, we also give a quantitative form of Khintchine's inequality in the special case $p=1$.
We investigate the R\'enyi entropy of independent sums of integer valued random variables through Fourier theoretic means, and give sharp comparisons between the variance and the R\'enyi entropy, for Poisson-Bernoulli variables. As applications we prove that a discrete ``min-entropy power'' is super additive on independent variables up to a universal constant, and give new bounds on an entropic generalization of the Littlewood-Offord problem that are sharp in the ``Poisson regime''.
We give a transport proof of a discrete version of the displacement convexity of entropy on integers (Z), and get, as a consequence, two discrete forms of the Pr{é}kopa-Leindler Inequality : the Four Functions Theorem of Ahlswede and Daykin on the discrete hypercube [1] and a recent result on Z due to Klartag and Lehec [16].
Motivated by Talagrand's conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and in particular its continuous analogue involving regularization properties of the Ornstein-Uhlenbeck semigroup acting on in-tegrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure.
We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete measures satisfying weak transport-entropy inequalities are also given.
We study an optimal weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities in dimension one. In particular, we obtain a weak transport-entropy form of the convex Poincar{é} inequality in dimension one.
In this note we bound the deficit in the logarithmic Sobolev Inequality and in the Talagrand transport-entropy Inequality for the Gaussian measure, in any dimension, by mean of a distance introduced by Bucur and Fragalà.
The de cit in the logarithmic Sobolev inequality for the Gaussian measure is considered and estimated by means of transport and information-theoretic distances.