Related papers: The Kannan-Lov\'asz-Simonovits Conjecture
We prove that if a triplet of functions satisfies almost equality in the Pr\'ekopa-Leindler inequality, then these functions are close to a common log-concave function, up to multiplication and rescaling. Our result holds for general…
The celebrated L\'evy--Khintchine theorem is a fundamental limiting law that describes the growth rate of the denominators of the convergents in the continued fraction expansion of a Lebesgue-typical real number. In a recent breakthrough,…
The purpose of this note is to explain that the combinatorial local log-concavity conjecture introduced by Gross, Mansour, Tucker and Wang (Eur. J. Comb. 52, 207-222, 2016) in fact follows from a result of Stanley (Eur. J. Comb. 32 (6),…
We consider the isoperimetric inequality on the class of high-dimensional isotropic convex bodies. We establish quantitative connections between two well-known open problems related to this inequality, namely, the thin shell conjecture, and…
The primary aim of this work is to develop methods that provide new insights into the relationships between fundamental constants in Banach space theory--specifically, the projection constant, the unconditional basis constant and the…
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…
By the Pr\'ekopa-Leindler inequality, the difference $X-X'$ has a log-concave density provided that $X$ has a log-concave density and $X, X'$ are independent and identically distributed. We prove that the opposite direction does not always…
Let $P$ be a finite poset with an element $s$ such that (1) for all $x\in P$, either $s\vee x$ or $s\wedge x$ exists; and (2) for all $x,y\in P$ such that $x<y$, if $s\wedge x$ does not exist but $s\wedge y$ does exist, then $(s\wedge…
In this article we review recent generalisations of the central limit theorem for the sum of specially correlated (or q-independent) variables, focusing on q greater or equal than 1. Specifically, this kind of correlation turns the…
The present note is a result of an on-going investigation into the logarithmic Brunn-Minkowski inequality. We obtain lower estimates on the volume product for convex bodies in $\mathbb{R}^n$ not necessarily symmetric with respect to the…
Lov\'asz Local Lemma (LLL) is a probabilistic tool that allows us to prove the existence of combinatorial objects in the cases when standard probabilistic argument does not work (there are many partly independent conditions). LLL can be…
In this paper, we extend the scope of Caffarelli's contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in $\R^d$. Our focus is on a broader category of…
It is well-known that measures whose density is the form $e^{-V}$ where $V$ is a uniformly convex potential on $\RR^n$ attain strong concentration properties. In search of a notion of log-concavity on the discrete hypercube, we consider…
We develop a novel stability theory for Sinkhorn semigroups based on Lyapunov techniques and quantitative contraction coefficients, and establish exponential convergence of Sinkhorn iterations on weighted Banach spaces. This…
The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank~1. This conjecture has been proved by Z. Szab\'{o} \cite{Sz} for harmonic manifolds with compact universal cover. E. Damek…
The log-concave maximum likelihood estimator of a density on the real line based on a sample of size $n$ is known to attain the minimax optimal rate of convergence of $O(n^{-4/5})$ with respect to, e.g., squared Hellinger distance. In this…
We examine the vertical component of surface area in the warped product of a Euclidean interval and a fiber manifold with product density. We determine general conditions under which vertical fibers minimize vertical surface area among…
Assuming the generalized continuum hypothesis we construct arbitrarily big indecomposable Banach spaces. i.e., such that whenever they are decomposed as $X\oplus Y$, then one of the closed subspaces $X$ or $Y$ must be finite dimensional. It…
This paper investigates a local central limit theorem for a normalized sequence of random variables belonging to a fixed order Wiener chaos and converging to the standard normal distribution. We prove, without imposing any additional…
Very recently, E. H. Lieb and J. P. Solovej stated a conjecture about the constant of embedding between two Bergman spaces of the upper-half plane. A question in relation with a Werhl-type entropy inequality for the affine $AX+B$ group.…