Bo'az Klartag, Roman Vershynin
Large deviation estimates are by now a standard tool inthe Asymptotic Convex Geometry, contrary to small deviationresults. In this note we present a novel application of a smalldeviations inequality to a problem related to the diameters of random sections of high dimensional convex bodies. Our results imply an unexpected distinction between the lower and the upper inclusions in Dvoretzky Theorem.
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A polynomial bound in Dvoretzky's theorem
本文给出ε-Dvoretzky猜想的简单概率证明，表明Dvoretzky定理中对逼近参数ε的依赖为多项式，并得到更尖锐的界及有限凸体族的同步版本，同时将逼近球半径与平均宽度等几何参数比较。
We study the dependence on $\varepsilon$ in the critical dimension $k(n,p,\varepsilon)$ for which one can find random sections of the $\ell_p^n$-ball which are $(1+\varepsilon)$-spherical. We give lower (and upper) estimates for $k(n,p,\varepsilon)$ for all eligible values $p$ and $\varepsilon$ as $n\to \infty$, which agree with the sharp estimates for the extreme values $p=1$ and $p=\infty$. Toward this end, we provide tight bounds for the Gaussian concentration of the $\ell_p$-norm.
Many star bodies have convex subsets with approximately the same Gaussian measure (of the complement). Inspired by this phenomenon, and in connection with the randomized Dvoretzky theorem for Lorentz spaces, we derive bounds on the distribution of certain functions of a Gaussian random vector by approximating their sub-level sets by convex subsets.
The isodiametric inequality states that the Euclidean ball maximizes the volume among all convex bodies of a given diameter. We are motivated by a conjecture of Makai Jr.~on the reverse question: Every convex body has a linear image whose isodiametric quotient is at least as large as that of a regular simplex. We relate this reverse isodiametric problem to minimal volume enclosing ellipsoids and to the Dvoretzky-Rogers-type problem of finding large volume simplices in any decomposition of the identity matrix. As a result, we solve the reverse isodiametric problem for $o$-symmetric convex bodie…
Given a class of functions $F$ on a probability space $(Ω,μ)$, we study the structure of a typical coordinate projection of the class, defined by $\{(f(X_i))_{i=1}^N : f \in F\}$, where $X_1,...,X_N$ are independent, selected according to $μ$. This notion of projection generalizes the standard linear random projection used in Asymptotic Geometric Analysis. We show that when $F$ is a subgaussian class of functions, a typical coordinate projection satisfies a Dvoretzky type theorem.
Dvoretzky-type theorem for locally finite subsets of a Hilbert space
本文证明了 $\ell_2$ 的任意局部有限子集可 $(1+\varepsilon)$-双 Lipschitz 嵌入到任意无限维 Banach 空间，并基于两个关于有限维空间直和与无条件基的独立结果。
We prove that for any $2<p<\infty$ and for every $n$-dimensional subspace $X$ of $L_p$, represented on $\mathbb R^n$, whose unit ball $B_X$ is in Lewis' position one has the following two-level Gaussian concentration inequality: \[ \mathbb P\left( \big| \|Z\| - \mathbb E\|Z\| \big| > \varepsilon \mathbb E\|Z\| \right) \leq C \exp \left (- c \min \left\{ α_p \varepsilon^2 n, (\varepsilon n)^{2/p} \right\} \right), \quad 0<\varepsilon<1 , \] where $Z$ is a standard $n$-dimensional Gaussian vectors, $α_p>0$ is a constant depending only on $p$ and $C,c>0$ are absolute constants. As a consequence w…
Radii of Euclidean sections of $\ell_p$-balls
研究$\ell_p$-球的欧几里得截面半径，探讨其在理论计算机科学中的应用及开放问题
A certain inequality conjectured by Vershynin is studied. It is proved that for any $n$-dimensional symmetric convex body $K$ with inradius $w$ and $γ_{n}(K) \leq 1/2$ there is $γ_{n}(sK) \leq (2s)^{w^{2}/4}γ_{n}(K)$ for any $s \in [0,1]$. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false.
Let $X$ be an isotropic random vector in $R^d$ that satisfies that for every $v \in S^{d-1}$, $\|<X,v>\|_{L_q} \leq L \|<X,v>\|_{L_p}$ for some $q \geq 2p$. We show that for $0<\varepsilon<1$, a set of $N = c(p,q,\varepsilon) d$ random points, selected independently according to $X$, can be used to construct a $1 \pm \varepsilon$ approximation of the $L_p$ unit ball endowed on $R^d$ by $X$. Moreover, $c(p,q,\varepsilon) \leq c^p \varepsilon^{-2}\log(2/\varepsilon)$; when $q=2p$ the approximation is achieved with probability at least $1-2\exp(-cN \varepsilon^2/\log^2(2/\varepsilon))$ and if $q$…
Small Ball Probabilities for Simple Random Tensors
本文研究由对数凹或坐标有界密度随机向量构成的简单随机张量的小球概率，给出投影落入小欧氏球的精确上界，并证明随机子空间情形下存在更优估计。
To prove by probabilistic methods that every $(n-1)$-dimensional section of the unit cube in $R^n$ has volume at most $\sqrt 2$, K. Ball made essential use of the inequality $$ \frac{1}π\int_{-\infty}^{\infty} \left(\frac{\sin^2 t}{t^2}\right)^pdt\leq \frac{\sqrt 2}{\sqrt p}, \quad p\geq 1, $$ in which equality holds if and only if $p=1$. The right side of above inequality has the correct rate of decay though the limit of the ratio of the right to left side is ${\sqrt{\frac{3}π}}$ rather then $\sqrt 2$. Applying Ball's methods we put all of this into the improved form of the Ball's inequality.
The slicing conjecture via small ball estimates
本研究借助随机局部化过程与Guan界，建立迷向对数凹测度的小球概率估计，给出Bourgain切片猜想的另一种证明，关联基于Milman的M-椭球理论。
Most likely balls in Banach spaces: existence and non-existence
本文提出 Banach 空间中固定形状凸集（如球）最大概率存在的一般准则，并给出反例说明其存在性在某些常见情形下可能不成立。
In this article we prove three fundamental types of limit theorems for the $q$-norm of random vectors chosen at random in an $\ell_p^n$-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guédon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of $\ell_p^n$-balls to lo…
Although there doesn't exist the Lebesgue measure in the ball $M$ of $C[0,1]$ with $p-$norm, the average values (expectation) $EY$ and variance $DY$ of some functionals $Y$ on $M$ can still be defined through the procedure of limitation from finite dimension to infinite dimension. In particular, the probability densities of coordinates of points in the ball $M$ exist and are derived out even though the density of points in $M$ doesn't exist. These densities include high order normal distribution, high order exponent distribution. This also can be considered as the geometrical origins of these …
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