相关论文: On some inequalities for Gaussian measures
This note presents families of inequalities for the Gaussian measure of convex sets which extend the recently proven Gaussian correlation inequality in various directions.
In this paper, we prove the isoperimetric inequality for the anisotropic Gaussian measure and characterize the cases of equality. We also find an example that shows Ehrhard symmetrization fails to decrease for the anisotropic Gaussian…
This paper is devoted to Gaussian interpolation inequalities with endpoint cases corresponding to the Gaussian Poincar\'e and the logarithmic Sobolev inequalities, seen as limits in large dimensions of Gagliardo-Nirenberg-Sobolev…
We discuss several classical and recent proofs of the isoperimetric inequality and the Sobolev inequality.
We give the counter-examples related to a Gaussian Brunn-Minkowski inequality and the (B) conjecture.
In this paper we present a correlation inequality with respect to Cauchy type measures. To prove our inequality, we transport the problem onto the Riemannian sphere then state and solve some special cases for a spherical correlation…
In the paper we investigate various inequalities for the one-dimensional Cauchy measure. We also consider analogous properties for one-dimensional sections of multidimensional isotropic Cauchy measure. The paper is a continuation of our…
Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev…
The paper is to prove the Gaussian correlation conjecture stating that, under the standard Gaussian measure, the measure of the intersection of any two symmetric convex sets is greater than or equal to the product of their measures.…
A positive correlation inequality is established for circular-invariant plurisubharmonic functions, with respect to complex Gaussian measures. The main ingredients of the proofs are the Ornstein-Uhlenbeck semigroup, and another natural…
We present isocapacitary characterizations of Sobolev inequalities in very general metric measure spaces.
The Ehrhard-Borell inequality is a far-reaching refinement of the classical Brunn-Minkowski inequality that captures the sharp convexity and isoperimetric properties of Gaussian measures. Unlike in the classical Brunn-Minkowski theory, the…
The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or…
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…
We prove some special cases of Bergeron's inequality involving two Gaussian polynomials (or $q$-binomials).
We give an "elementary" proof of an inequality due to Maz'ya. As a prerequisite we prove an approximation property for the Hausdorff measure. We also comment on the relations between Maz'ya's inequality, the isoperimetric inequality and the…
A stability version of the reverse isoperimetric inequality, and the corresponding inequality for isotropic measures are established.
The Gaussian product inequality is a long-standing conjecture. In this paper, we investigate the three-dimensional inequality $E[X_1^{2}X_2^{2m_2}X_3^{2m_3}]\ge E[X_1^{2}]E[X_2^{2m_2}]E[X_3^{2m_3}]$ for any centered Gaussian random vector…
The de cit in the logarithmic Sobolev inequality for the Gaussian measure is considered and estimated by means of transport and information-theoretic distances.
We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the…