A probabilistic approach to the geometry of the \ell_p^n-ball

We consider products of uniform random variables from the Stiefel manifold of orthonormal $k$-frames in $\mathbb{R}^n$, $k \le n$, and random vectors from the $n$-dimensional $\ell_p^n$-ball $\mathbb{B}_p^n$ with certain $p$-radial distributions, $p\in[1,\infty)$. The distribution of this product geometrically corresponds to the projection of the $p$-radial distribution on $\mathbb{B}^n_p$ onto a random $k$-dimensional subspace. We derive large deviation principles (LDPs) on ...

Recent advances in statistics introduced versions of the central limit theorem for high-dimensional vectors, allowing for the construction of confidence regions for high-dimensional parameters. In this note, $s$-sparsely convex high-dimensional confidence regions are compared with respect to their volume. Specific confidence regions which are based on $\ell_p$-balls are found to have exponentially smaller volume than the corresponding hypercube. The theoretical results are va...

For a random vector X in R^n, we obtain bounds on the size of a sample, for which the empirical p-th moments of linear functionals are close to the exact ones uniformly on an n-dimensional convex body K. We prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis. In particular, we find a short Lewis type decomposition for any finite dimensional subspace of L_p. We also prove that for an isotropic log-concave rand...

Let $K$ be an isotropic symmetric convex body in ${\mathbb R}^n$. We show that a subspace $F\in G_{n,n-k}$ of codimension $k=\gamma n$, where $\gamma\in (1/\sqrt{n},1)$, satisfies $$K\cap F\subseteq \frac{c}{\gamma }\sqrt{n}L_K (B_2^n\cap F)$$ with probability greater than $1-\exp (-\sqrt{n})$. Using a different method we study the same question for the $L_q$-centroid bodies $Z_q(\mu )$ of an isotropic log-concave probability measure $\mu $ on ${\mathbb R}^n$. For every $1\le...

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: \[ \mathbb P\left( \big| \|Z\| - \mathbb E\|Z\| \big| > \varepsilon \mathbb E\|Z\| \right) \leq C \exp \left (- c \min \left\{ \alpha_p \varepsilon^2 n, (\varepsilon n)^{2/p} \right\} \right), \quad 0<\varepsilon<1 , \] where $Z$ is a standard $n$-dim...

We extend classical estimates for the vector balancing constant of $\mathbb{R}^d$ equipped with the Euclidean and the maximum norms proved in the 1980's by showing that for $p =2$ and $p=\infty$, given vector families $V_1, \ldots, V_n \subset B_p^d$ with $0 \in \sum_{i=1}^n \mathrm{conv} \, V_i$, one may select vectors $v_i \in V_i$ with \[ \| v_1 + \ldots + v_n \|_2 \leq \sqrt{d} \] for $p=2$, and \[ \| v_1 + \ldots + v_n \|_\infty \leq O(\sqrt{d}) \] for $p = \infty$. Thes...

In this article we take a probabilistic look at H\"older's inequality, considering the ratio of terms in the classical H\"older inequality for random vectors in $\mathbb{R}^n$. We prove a central limit theorem for this ratio, which then allows us to reverse the inequality up to a multiplicative constant with high probability. The models of randomness include the uniform distribution on $\ell_p^n$ balls and spheres. We also provide a Berry-Esseen type result and prove a large ...

We derive tight non-asymptotic bounds for the Kolmogorov distance between the probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimension-free and depend on the nuclear (Schatten-one) norm of the difference between the covariance operators of the elements and on the norm of the mean shift. The obtained bounds significantly improve the bound based on Pinsker's inequality via the Kullback-Leibler divergenc...

We study the volume of unit balls $B^n_{p,q}$ of finite-dimensional Lorentz sequence spaces $\ell_{p,q}^n.$ We give an iterative formula for ${\rm vol}(B^n_{p,q})$ for the weak Lebesgue spaces with $q=\infty$ and explicit formulas for $q=1$ and $q=\infty.$ We derive asymptotic results for the $n$-th root of ${\rm vol}(B^n_{p,q})$ and show that $[{\rm vol}(B^n_{p,q})]^{1/n}\approx n^{-1/p}$ for all $0<p<\infty$ and $0<q\le\infty.$ We study further the ratio between the volume ...

Let $K$ be a smooth convex set with volume one in $\BBR^d$. Choose $n$ random points in $K$ independently according to the uniform distribution. The convex hull of these points, denoted by $K_n$, is called a {\it random polytope}. We prove that several key functionals of $K_n$ satisfy the central limit theorem as $n$ tends to infinity.