From finite- to infinite-dimensional phenomena in geometric functional analysis on local and asymptotic levels.
The moduli of smoothness and convexity and the Rademacher averages of the trace classes (1≤p<∞)
Finite-dimensional subspaces of uniformly convex and uniformly smooth Banach lattices and trace classes
On nearly euclidean decomposition for some classes of Banach spaces
Volumetric invariants and operators on random families of Banach spaces
The geometry of random projections of centrally symmetric convex bodies in is studied. It is shown that if for such a body K the Euclidean ball is the ellipsoid of minimal volume containing it and a random n-dimensional projection is “far” from then the (random) body B is as “rigid” as its “distance” to permits. The result holds for the full range of dimensions 1 ≤ n ≤ λN, for arbitrary λ ∈ (0,1).
Subspaces of ℓ₂(X) and Rad(X) without local unconditional structure
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.
Spaces with maximal projection constants
We show that n-dimensional spaces with maximal projection constants exist not only as subspaces of 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.
Essentially-Euclidean convex bodies
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...
Chevet type inequality and norms of submatrices
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 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...
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