Related papers: Sudakov-type minoration for log-concave vectors

We develop in this paper an amelioration of the method given by S. Bobkov and M. Ledoux in GAFA (2000). We prove by Prekopa-Leindler Theorem an optimal modified logarithmic Sobolev inequality adapted for all log-concave measure on $\dR^n$.…

In this paper we develop tools for studying limit theorems by means of convexity. We establish bounds for the discrepancy in total variation between probability measures $\mu$ and $\nu$ such that $\nu$ is log-concave with respect to $\mu$.…

We present a general subadditivity inequality for log-Sobolev constants of convolution measures. As a corollary, we show that the log-Sobolev constant is monotone along the sequence of standardized convolutions in the central limit theorem.

We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for…

We develop a general framework to study concavity properties of weighted marginals of $\beta$-concave functions on $\mathbb{R}^n$ via local methods. As a concrete implementation of our approach, we obtain a functional version of the…

We discuss a conjecture about comparability of weak and strong moments of log-concave random vectors and show the conjectured inequality for unconditional vectors in normed spaces with a bounded cotype constant.

We consider finite-state Markov chains that can be naturally decomposed into smaller ``projection'' and ``restriction'' chains. Possibly this decomposition will be inductive, in that the restriction chains will be smaller copies of the…

We derive two-sided bounds for expected values of suprema of canonical processes based on random variables with moments growing regularly. We also discuss a Sudakov-type minoration principle for canonical processes.

Shokurov conjectured that the set of all log canonical thresholds on varieties of bounded dimension satisfies the ascending chain condition. In this paper we prove that the conjecture holds for log canonical thresholds on smooth varieties…

We derive explicit bounds for the computation of normalizing constants $Z$ for log-concave densities $\pi = \exp(-U)/Z$ with respect to the Lebesgue measure on $\mathbb{R}^d$. Our approach relies on a Gaussian annealing combined with recent…

Assume that f is a strict convex function with a unique minimum in R^n. We divide the vector of n-variables to d groups of vector subvariables with d at least two. We assume that we can find the partial minimum of f with respect to each…

We prove new entropy inequalities for log concave and s-concave functions that strengthen and generalize recently established reverse log Sobolev and Poincare inequalities for such functions. This leads naturally to the concept of…

We formulate a generalization of Vojta's conjecture in terms of log pairs and variants of multiplier ideals. In this generalization, a variety is allowed to have singularities. It turns out that the generalized conjecture for a log pair is…

In this work we study the concentration properties of log-concave measures that are curved only on a subspace of directions. Proofs uses an adapted version of the stochastic localization process.

In the study of concentration properties of isotropic log-concave measures, it is often useful to first ensure that the measure has super-Gaussian marginals. To this end, a standard preprocessing step is to convolve with a Gaussian measure,…

We prove the ACC conjecture for local volumes. Moreover, when the local volume is bounded away from zero, we prove Shokurov's ACC conjecture for minimal log discrepancies.

We give counterexamples to Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients.

The Paouris inequality gives the large deviation estimate for Euclidean norms of log-concave vectors. We present a modified version of it and show how the new inequality may be applied to derive tail estimates of l_r-norms and suprema of…

We prove a Kahane-Khinchin type result with a few random vectors, which are distributed independently with respect to an arbitrary log-concave probability measure on $\R^n$. This is an application of small ball estimate and Chernoff's…

We review some simple techniques based on monotone mass transport that allow to obtain transport-type inequalities for any log-concave probability measure, and for more general measures as well. We discuss quantitative forms of these…