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### Directionally Euclidean structures of Banach spaces

Studia Mathematica

We study Banach spaces with directionally asymptotically controlled ellipsoid-approximations of the unit ball in finite-dimensional sections. Here these ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding finite-dimensional subspace. The directional control here means that we evaluate the ellipsoids by means of a given functional of the dual space. The term 'asymptotical' refers to the fact that we take 'lim sup' over finite-dimensional subspaces. ...

### Essentially-Euclidean convex bodies

Studia Mathematica

In this note we introduce a notion of essentially-Euclidean normed spaces (and convex bodies). Roughly speaking, an n-dimensional space is λ-essentially-Euclidean (with 0 < λ < 1) if it has a [λn]-dimensional subspace which has further proportional-dimensional Euclidean subspaces of any proportion. We consider a space X₁ = (ℝⁿ,||·||₁) with the property that if a space X₂ = (ℝⁿ,||·||₂) is "not too far" from X₁ then there exists a [λn]-dimensional subspace E⊂ ℝⁿ such that E₁ = (E,||·||₁) and...

### Euclidean arrangements in Banach spaces

Studia Mathematica

We study the way in which the Euclidean subspaces of a Banach space fit together, somewhat in the spirit of the Kashin decomposition. The main tool that we introduce is an estimate regarding the convex hull of a convex body in John's position with a Euclidean ball of a given radius, which leads to a new and simplified proof of the randomized isomorphic Dvoretzky theorem. Our results also include a characterization of spaces with nontrivial cotype in terms of arrangements of Euclidean subspaces.

### Generalizing the Johnson-Lindenstrauss lemma to k-dimensional affine subspaces

Studia Mathematica

Let ε > 0 and 1 ≤ k ≤ n and let ${{W}_{l}}_{l=1}^{p}$ be affine subspaces of ℝⁿ, each of dimension at most k. Let $m=O\left({\epsilon }^{-2}\left(k+logp\right)\right)$ if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map $H:ℝⁿ\to {ℝ}^{m}$ such that for all 1 ≤ l ≤ p and $x,y\in {W}_{l}$ we have ||x-y||₂ ≤ ||H(x)-H(y)||₂ ≤ (1+ε)||x-y||₂, i.e. the distance distortion is at most 1 + ε. The estimate on m is tight in terms of k and p whenever ε < 1, and is tight on ε,k,p whenever ε ≥ 1. We extend these results to embeddings into general normed spaces Y.

### Logconcave random graphs.

The Electronic Journal of Combinatorics [electronic only]

Studia Mathematica

We study the closures of classes of log-concave measures under taking weak limits, linear transformations and tensor products. We investigate which uniform measures on convex bodies can be obtained starting from some class 𝒦. In particular we prove that if one starts from one-dimensional log-concave measures, one obtains no non-trivial uniform mesures on convex bodies.

### On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

Studia Mathematica

We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix $H{ₙ}^{\left(0\right)}$ and of mₙ rank-one matrices determined by i.i.d. isotropic random vectors with log-concave probability law and real amplitudes. This is an analog of the setting of Marchenko and Pastur [Mat. Sb. 72 (1967)]. We prove that if mₙ/n → c ∈ [0,∞) as n → ∞, and the distribution of eigenvalues of $H{ₙ}^{\left(0\right)}$ and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges...

### Sudakov-type minoration for log-concave vectors

Studia Mathematica

We formulate and discuss a conjecture concerning lower bounds for norms of log-concave vectors, which generalizes the classical Sudakov minoration principle for Gaussian vectors. We show that the conjecture holds for some special classes of log-concave measures and some weaker forms of it are satisfied in the general case. We also present some applications based on chaining techniques.

### Thin-shell concentration for convex measures

Studia Mathematica

We prove that for s < 0, s-concave measures on ℝⁿ exhibit thin-shell concentration similar to the log-concave case. This leads to a Berry-Esseen type estimate for most of their one-dimensional marginal distributions. We also establish sharp reverse Hölder inequalities for s-concave measures.

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