On certain equivalent norms on Tsirelson's space

Tsirelson's space $T$ is known to be distortable but it is open as to whether or not $T$ is arbitrarily distortable. For $n\in {\Bbb N}$ the norm $\|\cdot\|_n$ of the Tsirelson space $T(S_n,2^{-n})$ is equivalent to the standard norm on $T$. We prove there exists $K<\infty$ so that for all $n$, $\|\cdot\|_n$ does not $K$ distort any subspace $Y$ of $T$.Comment: 19 pp., LaTe

Norms of Minimal Projections

It is proved that the projection constants of two- and three-dimensional spaces are bounded by $4/3$ and $(1+\sqrt 5)/2$, respectively. These bounds are attained precisely by the spaces whose unit balls are the regular hexagon and dodecahedron. In fact, a general inequality for the projection constant of a real or complex $n$-dimensional space is obtained and the question of equality therein is discussed

Proximity to ℓ1\ell_1 and Distortion in Asymptotic ℓ1\ell_1 Spaces

For an asymptotic $\ell_1$ space $X$ with a basis $(x_i)$ certain asymptotic $\ell_1$ constants, $\delta_\alpha (X)$ are defined for $\alpha <\omega_1$. $\delta_\alpha (X)$ measures the equivalence between all normalized block bases $(y_i)_{i=1}^k$ of $(x_i)$ which are $S_\alpha$-admissible with respect to $(x_i)$ ($S_\alpha$ is the $\alpha^{th}$-Schreier class of sets) and the unit vector basis of $\ell_1^k$. This leads to the concept of the delta spectrum of $X$, $\Delta (X)$, which reflects the behavior of stabilized limits of $\delta_\alpha (X)$. The analogues of these constants under all renormings of $X$ are also defined and studied. We investigate $\Delta (X)$ both in general and for spaces of bounded distortion. We also prove several results on distorting the classical Tsirelson's space $T$ and its relatives

Uniform uncertainty principle for Bernoulli and subgaussian ensembles

We present a simple solution to a question posed by Candes, Romberg and Tao on the uniform uncertainty principle for Bernoulli random matrices. More precisely, we show that a rectangular k*n random subgaussian matrix (with k < n) has the property that by arbitrarily extracting any m (with m < k) columns, the resulting submatrices are arbitrarily close to (multiples of) isometries of a Euclidean space. We obtain the optimal estimate for m as a function of k,n and the degree of "closeness" to an isometry. We also give a short and self-contained solution of the reconstruction problem for sparse vectors.Comment: 15 pages; no figures; submitte

On approximations by projections of polytopes with few facets

We provide an affirmative answer to a problem posed by Barvinok and Veomett, showing that in general an n-dimensional convex body cannot be approximated by a projection of a section of a simplex of a sub-exponential dimension. Moreover, we establish a lower bound of the Banach-Mazur distance between n-dimensional projections of sections of an N-dimensional simplex and a certain convex symmetric body, which is sharp up to a logarithmic factor for all N>n.Comment: 22 page

Structural properties of weak cotype 2 spaces

Several characterizations of weak cotype 2 and weak Hilbert spaces are given in terms of basis constants and other structural invariants of Banach spaces. For finite-dimensional spaces, characterizations depending on subspaces of fixed proportional dimension are proved

On the structure of the spreading models of a Banach space

We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space $X$. In particular we give an example of a reflexive $X$ so that all spreading models of $X$ contain $\ell_1$ but none of them is isomorphic to $\ell_1$. We also prove that for any countable set $C$ of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of $C$. In certain cases this ensures that $X$ admits, for each $\alpha < \omega_1$, a spreading model $(\tilde x_i^\alpha)_i$ such that if $\alpha < \beta$ then $(\tilde x_i^\alpha)_i$ is dominated by (and not equivalent to) $(\tilde x_i^\beta)_i$. Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map

Erratum to: ``Banach spaces without local unconditional structure''

This note contains a corrected proof of the main result (which remains unchanged) from [K-T]. It was recently observed that an argument in a basic technical criterium has a gap

Banach spaces without local unconditional structure

For a large class of Banach spaces, a general construction of subspaces without local unconditional structure is presented. As an application it is shown that every Banach space of finite cotype contains either $l_2$ or a subspace without unconditional basis, which admits a Schauder basis. Some other interesting applications and corollaries follow

On the interval of fluctuation of the singular values of random matrices

Let $A$ be a matrix whose columns $X_1,\dots, X_N$ are independent random vectors in $\mathbb{R}^n$. Assume that the tails of the 1-dimensional marginals decay as $\mathbb{P}(|\langle X_i, a\rangle|\geq t)\leq t^{-p}$ uniformly in $a\in S^{n-1}$ and $i\leq N$. Then for $p>4$ we prove that with high probability $A/{\sqrt{n}}$ has the Restricted Isometry Property (RIP) provided that Euclidean norms $|X_i|$ are concentrated around $\sqrt{n}$. We also show that the covariance matrix is well approximated by the empirical covariance matrix and establish corresponding quantitative estimates on the rate of convergence in terms of the ratio $n/N$. Moreover, we obtain sharp bounds for both problems when the decay is of the type $\exp({-t^{\alpha}})$ with $\alpha \in (0,2]$, extending the known case $\alpha\in[1, 2]$.Comment: To appear in J. Eur. Math. So