This article investigates, by probabilistic methods, various geometric questions on B_p^n, the unit ball of \ell_p^n. We propose realizations in terms of independent random variables of several distributions on B_p^n, including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in B_p^n. As another application, we compute moments of linear functionals on B_p^n, which gives sharp constants in Khinchine's inequalities on B_p^n and determines the ψ_2-constant of all directions on B_p^n. We also study the extremal values of several Gaussian averages on sections of B_p^n (including mean width and \ell-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in \ell_2 and to covering numbers of polyhedra complete the exposition.
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 the space of probability measures on $\mathbb{R}^k$ for sequences of such projections.
Sharp large deviation results of Bahadur-Ranga Rao type are provided for the $q$-norm of random vectors distributed on the $\ell _{p}^{n}$-ball ${\mathbb{B}}^{n}_{p}$ according to the cone probability measure or the uniform distribution for $1 \le q<p < \infty $, thereby furthering previous large deviation results by Kabluchko, Prochno and Thäle in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different $\ell _{p}^{n}$-balls in the spirit of Schechtman and Schmuckenschläger, and for the length of the projection of an $\ell _{p}^{n}$-ba…
We show that the two-dimensional minimum-volume central section of the $n$-dimensional cross-polytope is attained by the regular $2n$-gon. We establish stability-type results for hyperplane sections of $\ell_p$-balls in all the cases where the extremisers are known. Our methods are mainly probabilistic, exploring connections between negative moments of projections of random vectors uniformly distributed on convex bodies and volume of their sections.
Let $p\in[1,\infty]$. Consider the projection of a uniform random vector from a suitably normalized $\ell^p$ ball in $\mathbb{R}^n$ onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension $n$ goes to $\infty$, which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for $p\in(1,\infty]$ (but not for $p=1$), the corresponding rate function is "universal", in the sense that …
In this paper, we prove a multivariate central limit theorem for $\ell_q$-norms of high-dimensional random vectors that are chosen uniformly at random in an $\ell_p^n$-ball. As a consequence, we provide several applications on the intersections of $\ell_p^n$-balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an $\ell_p^n$-ball onto a line spanned by a random direction $θ\in\mathbb S^{n-1}$. The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreov…
The normalised volume measure on the $\ell_p^n$ unit ball ($1\leq p\leq 2$) satisfies the following isoperimetric inequality: the boundary measure of a set of measure $a$ is at least $cn^{1/p}\tilde{a}\log^{1-1/p}(1/\tilde{a})$, where $\tilde{a}=\min(a,1-a)$.
The aims of this paper are twofold. Firstly, we derive some probabilistic representation for the constant which appears in the one-dimensional case of Kesten's renewal theorem. Secondly, we estimate the tail of some related random variable which plays an essential role in the description of the stable limit law of one-dimensional transient sub-ballistic random walks in random environment.
In this work we study the rate of convergence in the central limit theorem for the Euclidean norm of random orthogonal projections of vectors chosen at random from an $\ell_p^n$-ball which has been obtained in [Alonso-Gutiérrez, Prochno, Thäle: Gaussian fluctuations for high-dimensional random projections of $\ell_p^n$-balls, Bernoulli 25(4A), 2019, 3139--3174]. More precisely, for any $n\in\mathbb N$ let $E_n$ be a random subspace of dimension $k_n\in\{1,\ldots,n\}$, $P_{E_n}$ the orthogonal projection onto $E_n$, and $X_n$ be a random point in the unit ball of $\ell_p^n$. We prove a Berry-Es…
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…
A question in geometric probability about the location of the balls in a game of bocce leads to related questions about the probability that a system of linear equations has a positive solution and the probability that a random zero-sum game favors the row player. Under reasonable assumptions we are able to find these probabilities.
In this paper, we study high-dimensional random projections of $\ell_p^n$-balls. More precisely, for any $n\in\mathbb N$ let $E_n$ be a random subspace of dimension $k_n\in\{1,\ldots,n\}$ and $X_n$ be a random point in the unit ball of $\ell_p^n$. Our work provides a description of the Gaussian fluctuations of the Euclidean norm $\|P_{E_n}X_n\|_2$ of random orthogonal projections of $X_n$ onto $E_n$. In particular, under the condition that $k_n\to\infty$ it is shown that these random variables satisfy a central limit theorem, as the space dimension $n$ tends to infinity. Moreover, if $k_n\to\i…
The paper provides a description of the large deviation behavior for the Euclidean norm of projections of $\ell_p^n$-balls to high-dimensional random subspaces. More precisely, for each integer $n\geq 1$, let $k_n\in\{1,\ldots,n-1\}$, $E^{(n)}$ be a uniform random $k_n$-dimensional subspace of $\mathbb R^n$ and $X^{(n)}$ be a random point that is uniformly distributed in the $\ell_p^n$-ball of $\mathbb R^n$ for some $p\in[1,\infty]$. Then the Euclidean norms $\|P_{E^{(n)}}X^{(n)}\|_2$ of the orthogonal projections are shown to satisfy a large deviation principle as the space dimension $n$ tend…
We establish Central Limit Theorems for the volumes of intersections of $B_{p}^n$ (the unit ball of $\ell_p^n$) with uniform random subspaces of codimension $d$ for fixed $d$ and $n\to \infty$. As a corollary we obtain higher order approximations for expected volumes, refining previous results by Koldobsky and Lifschitz and approximations obtained from the Eldan--Klartag version of CLT for convex bodies. We also obtain a Central Limit Theorem for the Minkowski functional of the intersection body of $B_p^n$, evaluated on a random vector distributed uniformly on the unit sphere.
In this article we take a probabilistic look at Hölder's inequality, considering the ratio of terms in the classical Hölder 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 and a moderate deviation principle for the suitably normalized Hölder ratio.
Let $Q_{n}^{r}$ be the graph with vertex set $\{-1,1\}^{n}$ in which two vertices are joined if their Hamming distance is at most $r$. The edge-isoperimetric problem for $Q_{n}^{r}$ is that: For every $(n,r,M)$ such that $1\le r\le n$ and $1\le M\le2^{n}$, determine the minimum edge-boundary size of a subset of vertices of $Q_{n}^{r}$ with a given size $M$. In this paper, we apply two different approaches to prove bounds for this problem. The first approach is a linear programming approach and the second is a probabilistic approach. Our bound derived by the first approach generalizes the tight…
We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of the ambient space tends to infinity. This generalizes a result of Schechtman and Schmuckenschläger [GAFA, Lecture notes in Math. 1469 (1991), 174--178] for $\ell_p^d$-balls. As another application, we determine the precise asymptotic volume ratio for $2$-concave Orlicz spaces $\ell_M^d$. Our method rests on ideas from statistical mechanics and large deviations theory, more precisely the maximum entropy or Gibbs prin…
Let $u$ be a pluriharmonic function on the unit ball in $\mathbb{C}^n$. I consider the relationship between the set of points $L_u$ on the boundary of the ball at which $u$ converges nontangentially and the set of points $\mathcal{L}_u$ at which $u$ converges along conditioned Brownian paths. For harmonic functions $u$ of two variables, the result $L_u\stackrel{\mathrm{a.e.}}{=}\mathcal{L}_u$ has been known for some time, as has a counterexample to the same equality for three variable harmonic functions. I extend the $L_u\stackrel{\mathrm{a.e.}}{=}\mathcal{L}_u$ result to pluriharmonic functio…
