Related papers: The Gaussian Conjugate Rogers-Shephard Inequality

In this manuscript, we study the inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if convex bodies $K, L$ satisfy $|K|\theta^{\perp}|…

We consider a family of jointly Gaussian random vectors $\xi_j \in \mathbb{R}^{k_j}$, each standard normal but possibly correlated, and investigate when\[ \mathbb{E}\, F\!\Bigl(B\bigl(|T_{z_1} f_1(\xi_1)|,\dots,|T_{z_n}…

In this paper we investigate the reverse isoperimetric inequality with respect to the Gaussian measure for convex sets in $\mathbb{R}^{2}$. While the isoperimetric problem for the Gaussian measure is well understood, many relevant aspects…

A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds…

The Gaussian Correlation Conjecture states that for any two symmetric, convex sets in n-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is greater than or equal to the product of the…

The classical Gaussian concentration inequality for Lipschitz functions is adapted to a setting where the classical assumptions (i.e. Lipschitz and Gaussian) are not met. The theory is more direct than much of the existing theory designed…

In this paper we address the following question: given a measure $\mu$ on $\mathbb{R}^n$, does there exists a constant $C>0$ such that, for any $m$-dimensional subspace $H \subset \mathbb{R}^n$ and any convex body $K \subset \mathbb{R}^n$,…

The long-standing Godbersen's conjecture asserts that the Rogers-Shephard inequality for the volume of the difference body is refined by an inequality for the mixed volume of a convex body and its reflection about the origin. The conjecture…

The Riesz-Sobolev inequality provides an upper bound, in integral form, for the convolution of indicator functions of subsets of Euclidean space. We formulate and prove a sharper form of the inequality. This can be equivalently phrased as a…

In the context of mod-Gaussian convergence, as defined previously in our work with J. Jacod, we obtain lower bounds for local probabilities for a sequence of random vectors which are approximately Gaussian with increasing covariance. This…

We revisit Royen's proof of the Gaussian correlation inequality from a supersymmetric point of view. Many key elements in Royen's proof of this inequality have natural geometric interpretations in terms of supersymmetric dimensional…

Let K \subset R^N be a convex body containing the origin. A measurable set G \subset R^N with positive Lebesgue measure is said to be uniformly K-dense if, for any fixed r > 0, the measure of G \cap (x + rK) is constant when x varies on the…

We prove stability estimates for the Bakry-Emery bound on Poincar\'e and logarithmic Sobolev constants of uniformly log-concave measures. In particular, we improve the quantitative bound in a result of De Philippis and Figalli asserting…

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…

Given even strongly log-concave random vectors $X_{0}$ and $X_{1}$ in $\mathbb{R}^n$, we show that a natural joint distribution $(X_{0},X_{1})$ satisfies, \begin{equation} e^{ - \frac{1}{n}D ((1-t)X_{0} + t X_{1} \Vert Z)} \geq (1-t) e^{ -…

Given a probability measure $\mu$ supported on a convex subset $\Omega$ of Euclidean space $(\mathbb{R}^d,g_0)$, we are interested in obtaining Poincar\'e and log-Sobolev type inequalities on $(\Omega,g_0,\mu)$. To this end, we change the…

In this article, we derive a new covariance estimate. The estimate has a similar structure as the Brascamp-Lieb inequality and is optimal for ferromagnetic Gaussian measures. It can be naturally applied to deduce decay of correlations of…

A radial probability measure is a probability measure with a density (with respect to the Lebesgue measure) which depends only on the distances to the origin. Consider the Euclidean space enhanced with a radial probability measure. A…

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…

Estimates are obtained for the Lebesgue constants associated with the Gauss quadrature points on $(-1, +1)$ augmented by the point $-1$ and with the Radau quadrature points on either $(-1, +1]$ or $[-1, +1)$. It is shown that the Lebesgue…