相关论文: Small ball probability and Dvoretzky theorem
The work of Gantert, Kim, and Ramanan [Large deviations for random projections of $\ell^p$ balls, Ann. Probab. 45 (6B), 2017] has initiated and inspired a new direction of research in the asymptotic theory of geometric functional analysis.…
A method is introduced for studying large deviations in the context of statistical physics of disordered systems. The approach, based on an extension of the cavity method to atypical realizations of the quenched disorder, allows us to…
In the present paper, the following convexity principle is proved: any closed convex multifunction, which is metrically regular in a certain uniform sense near a given point, carries small balls centered at that point to convex sets, even…
For a $d$-dimensional random vector $X$, let $p_{n, X}(\theta)$ be the probability that the convex hull of $n$ independent copies of $X$ contains a given point $\theta$. We provide several sharp inequalities regarding $p_{n, X}(\theta)$ and…
In this short note, we hope to give a rapid induction for non-experts into the world of Differential Harnack inequalities, which have been so influential in geometric analysis and probability theory over the past few decades. At the…
Existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of random rotations of K and L is nicely bounded. For L = subspace, this main result immediately yields the unexpected phenomenon: "If…
We consider high-dimensional estimation problems where the number of parameters diverges with the sample size. General conditions are established for consistency, uniqueness, and asymptotic normality in both unpenalized and penalized…
This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new direction in geometric tomography where different…
The geometry of unit $N$-dimensional $\ell_{p}$ balls has been intensively investigated in the past decades. A particular topic of interest has been the study of the asymptotics of their projections. Apart from their intrinsic interest,…
The famous Minkowski inequality provides a sharp lower bound for the mixed volume $V(K,M[n-1])$ of two convex bodies $K,M\subset\mathbb{R}^n$ in terms of powers of the volumes of the individual bodies $K$ and $M$. The special case where $K$…
The isoperimetric problem is a classic topic in geometric measure theory, yet critical questions regarding the characterization of optimal solutions -- even asymptotically optimal ones -- remain largely unresolved. In this paper, we…
Random matrices have played an important role in many fields including machine learning, quantum information theory and optimization. One of the main research focuses is on the deviation inequalities for eigenvalues of random matrices.…
This note contains two types of small ball estimates for random vectors in finite dimensional spaces equipped with a quasi-norm. In the first part, we obtain bounds for the small ball probability of random vectors under some smoothness…
We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given…
The variance conjecture in Asymptotic Convex Geometry stipulates that the Euclidean norm of a random vector uniformly distributed in a (properly normalised) high-dimensional convex body $K\subset {\mathbb R}^n$ satisfies a Poincar\'e-type…
The Small Ball Inequality is a conjectural lower bound on sums the L-infinity norm of sums of Haar functions supported on dyadic rectangles of a fixed volume in the unit cube. The conjecture is fundamental to questions in discrepancy…
In this survey, we discuss volumetric and combinatorial results concerning (mostly finite) intersections or unions of balls (mostly of equal radii) in the $d$-dimensional real vector space, mostly equipped with the Euclidean norm. Our first…
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…
Given an Euclidean space, this paper elucidates the topological link between the partial derivatives of the Minkowski functional associated to a set (assumed to be compact, convex, with a differentiable boundary and a non-empty interior)…
We revisit an ingenious argument of K. Ball to provide sharp estimates for the volume of sections of a convex body in John's position. Our technique combines the geometric Brascamp-Lieb inequality with a generalised Parseval-type identity.…