Related papers: Almost minimal orthogonal projections

We consider the problem of embedding a finite set of points $\{x_1, \ldots, x_n\} \in \mathbb{R}^d$ that satisfy $\ell_2^2$ triangle inequalities into $\ell_1$, when the points are approximately low-dimensional. Goemans (unpublished,…

We study a property of $2$-strong uniqueness of a best approximation in a class of finite-dimensional complex normed spaces, for which the unit ball is an absolutely convex hull of finite number of points and in its dual class. We prove…

Let {\mathbb{V} = V x R^l : V \in G(n-l,m-l)} be the family of m-dimensional subspaces of R^n containing {0} x R^l, and let \pi_{\mathbb{V}} : R^n --> \mathbb{V} be the orthogonal projection onto \mathbb{V}. We prove that the mapping V…

We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it…

We study the problem of $2$-dimensional orthogonal range counting with additive error. Given a set $P$ of $n$ points drawn from an $n\times n$ grid and an error parameter $\eps$, the goal is to build a data structure, such that for any…

The \emph{relative projection constant} $\lambda(Y, X)$ of normed spaces $Y \subset X$ is defined as $\lambda(Y, X) = \inf \{ ||P|| : P \in \mathcal{P}(X, Y) \}$, where $\mathcal{P}(X, Y)$ denotes the set of all continuous projections from…

In this paper we survey some results on minimality of projections with respect to numerical radius. We note that in the cases $L^p$, $p=1,2,\infty$, there is no difference between the minimality of projections measured either with respect…

Let $E$ be a Banach space and, for any positive integer $n$, let ${\cal P}(^nE)$ denote the Banach space of continuous $n$-homogeneous polynomials on $E$. Davie and Gamelin showed that the natural extension mapping from ${\cal P}(^nE)$ to…

Over the real or complex field, we establish a duality formula for projection constants of finite-codimensional subspaces of Banach spaces with the Daugavet property. If \[ Y=\bigcap_{j=1}^n \ker f_j \subset X, \qquad…

We consider Laplacian eigenfunctions on a $d-$dimensional bounded domain $M$ (or a $d-$dimensional compact manifold $M$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $(e_\ell)_{\ell \in \mathbb{N}}$.…

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 $k_n\in\{1,\ldots,n-1\}$,…

We prove the following proposition. Under linear interpolation on a Euclidean $n$-dimensional ball $B$, an interpolation projector whose nodes coincide with the vertices of a regular simplex inscribed into the boundary sphere has the…

Let $X$ be a given Banach space and let $M$, $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. The paper contains the proof of the estimate $$ \mathbb E \Psi(N_t) \leq…

Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric…

In a previous work we proved that if a finite Borel measure $\mu$ in a Euclidean space has Hausdorff dimension smaller than a positive integer $k$, then the orthogonal projection onto almost every $k$-dimensional linear subspace is…

For a Banach space X we define RUMD_n(X) to be the infimum of all c>0 such that (AVE_{\epsilon_k =\pm 1} || \sum_1^n epsilon_k (M_k - M_{k-1} )||_{L_2^X}^2 )^{1/2} <= c || M_n ||_{L_2^X} holds for all Walsh-Paley martingales {M_k}_0^n…

Low-distortion embeddings are critical building blocks for developing random sampling and random projection algorithms for linear algebra problems. We show that, given a matrix $A \in \R^{n \times d}$ with $n \gg d$ and a $p \in [1, 2)$,…

Let $\mu$ be a planar self-similar measure with similarity dimension exceeding $1$, satisfying a mild separation condition, and such that the fixed points of the associated similitudes do not share a common line. Then, we prove that the…

An oblivious subspace embedding (OSE), characterized by parameters $m,n,d,\epsilon,\delta$, is a random matrix $\Pi\in \mathbb{R}^{m\times n}$ such that for any $d$-dimensional subspace $T\subseteq \mathbb{R}^n$, $\Pr_\Pi[\forall x\in T,…

The main result of the paper: Given any $\varepsilon>0$, every locally finite subset of $\ell_2$ admits a $(1+\varepsilon)$-bilipschitz embedding into an arbitrary infinite-dimensional Banach space. The result is based on two results which…