Related papers: On Convex Functions of Gaussian Variables
We consider the well-travelled problem of homogenization of random integral functionals. When the integrand has standard growth conditions, the qualitative theory is well-understood. When it comes to unbounded functionals, that is, when the…
We provide a sharp rate of convergence in the central limit theorem for random vectors with an unconditional, log-concave density. The argument relies on analysis of the Neumann laplacian on convex domains and on the theory of optimal…
We introduce floating bodies for convex, not necessarily bounded subsets of $\mathbb{R}^n$. This allows us to define floating functions for convex and log concave functions and log concave measures. We establish the asymptotic behavior of…
In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of…
This paper studies the strong quasiconvexity of norm and distance functions in finite-dimensional normed spaces. Although the Euclidean norm is known to be strongly quasiconvex on bounded convex sets, a complete characterization of this…
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…
Recently, Bo'az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag's quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies…
We will prove a reverse Rogers-Shephard inequality for log-concave functions. In some particular cases, the method used for general log-concave functions can be slightly improved, allowing us to prove volume estimates for polars of…
We propose flexible Gaussian representations for conditional cumulative distribution functions and give a concave likelihood criterion for their estimation. Optimal representations satisfy the monotonicity property of conditional cumulative…
We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the $\mathrm{U}$-functional of a convex…
We consider nonconvex real valued functions whose truncations are either quasiconvex or even convex starting with a certain level. Among them, the $C^2$-smooth functions whose level sets are all completely contained in the positive definite…
The central limit theorem ensures that a sum of random variables tends to a Gaussian distribution as their total number tends to infinity. However, for a class of positive random variables, we find that the sum tends faster to a log-normal…
Motivated by connections to random matrices, Littlewood-Richardson coefficients and tilings, we study random discrete concave functions on an equilateral lattice. We show that such functions having a periodic Hessian of a fixed average…
This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide explicit…
We revisit the problem of bounding the expected supremum of a canonical Gaussian process indexed by a convex set $T \subset \mathbf{R}^d$. We develop two decompositions for the Gaussian width, based on the geometry of the index set. The…
The purpose of this paper is to provide a set of sufficient conditions so that the normalized form of the Fox-Wright functions have certain geometric properties like close-to-convexity, univalency, convexity and starlikeness inside the unit…
The restricted strong convexity is an effective tool for deriving globally linear convergence rates of descent methods in convex minimization. Recently, the global error bound and quadratic growth properties appeared as new competitors. In…
The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to…
A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures…
In this paper, we generalize and improve some fundamental concentration inequalities using information on the random variables' higher moments. In particular, we improve the classical Hoeffding's and Bennett's inequalities for the case…