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1
Content available remote

Subspaces of ℓ₂(X) and Rad(X) without local unconditional structure

100%
Komorowski R., Tomczak-Jaegermann N.
Studia Mathematica
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2002
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tom 149
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nr 1
1-21
EN
It is shown that if a Banach space X is not isomorphic to a Hilbert space then the spaces ℓ₂(X) and Rad(X) contain a subspace Z without local unconditional structure, and therefore without an unconditional basis. Moreover, if X is of cotype r < ∞, then a subspace Z of ℓ₂(X) can be constructed without local unconditional structure but with 2-dimensional unconditional decomposition, hence also with basis.
2
Content available remote

Volumetric invariants and operators on random families of Banach spaces

100%
Mankiewicz P., Tomczak-Jaegermann N.
Studia Mathematica
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2003
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tom 159
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nr 2
315-335
EN
The geometry of random projections of centrally symmetric convex bodies in $ℝ^{N}$ is studied. It is shown that if for such a body K the Euclidean ball $B₂^{N}$ is the ellipsoid of minimal volume containing it and a random n-dimensional projection $B = P_{H}(K)$ is "far" from $P_{H}(B₂^{N})$ then the (random) body B is as "rigid" as its "distance" to $P_{H}(B₂^{N})$ permits. The result holds for the full range of dimensions 1 ≤ n ≤ λN, for arbitrary λ ∈ (0,1).
3
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Spaces with maximal projection constants

100%
König H., Tomczak-Jaegermann N.
Studia Mathematica
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2003
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tom 159
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nr 3
357-372
EN
We show that n-dimensional spaces with maximal projection constants exist not only as subspaces of $l_{∞}$ but also as subspaces of l₁. They are characterized by a rigid set of vector conditions. Nevertheless, we show that, in general, there are many non-isometric spaces with maximal projection constants. Several examples are discussed in detail.
4
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Finite-dimensional subspaces of uniformly convex and uniformly smooth Banach lattices and trace classes $S_{p}$

99%
Tomczak-Jaegermann N.
Studia Mathematica
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1979-1980
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tom 66
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nr 3
261-281
5
Content available remote

The moduli of smoothness and convexity and the Rademacher averages of the trace classes $S_{p}$ (1≤p<∞)

99%
Tomczak-Jaegermann N.
Studia Mathematica
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1974
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tom 50
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nr 2
163-182
6
Content available remote

Essentially-Euclidean convex bodies

81%
Litvak A., Milman V., Tomczak-Jaegermann N.
Studia Mathematica
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2010
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tom 196
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nr 3
207-221
EN
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 E₂ = (E,||·||₂) are "very close." We then show that such an X₁ is λ-essentially-Euclidean (with λ depending only on quantitative parameters measuring "closeness" of two normed spaces). This gives a very strong negative answer to an old question of the second named author. It also clarifies a previously obtained answer by Bourgain and Tzafriri. We prove a number of other results of a similar nature. Our work shows that, in a sense, most constructions of the asymptotic theory of normed spaces cannot be extended beyond essentially-Euclidean spaces.
7
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Chevet type inequality and norms of submatrices

71%
Adamczak R., Latała R., Litvak A., Pajor A., Tomczak-Jaegermann N.
Studia Mathematica
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2012
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tom 210
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nr 1
35-56
EN
We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of the expectation of the supremum of "symmetric exponential" processes, compared to the Gaussian ones in the Chevet inequality. This is used to give a sharp upper estimate for a quantity $Γ_{k,m}$ that controls uniformly the Euclidean operator norm of the submatrices with k rows and m columns of an isotropic log-concave unconditional random matrix. We apply these estimates to give a sharp bound for the restricted isometry constant of a random matrix with independent log-concave unconditional rows. We also show that our Chevet type inequality does not extend to general isotropic log-concave random matrices.
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