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An isomorphic Dvoretzky's theorem for convex bodies

100%

Gordon Y., Guédon O., Meyer M.

Studia Mathematica

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1998

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tom 127

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nr 2

191-200

EN

We prove that there exist constants C>0 and 0 < λ < 1 so that for all convex bodies K in $ℝ^n$ with non-empty interior and all integers k so that 1 ≤ k ≤ λn/ln(n+1), there exists a k-dimensional affine subspace Y of $ℝ^n$ satisfying $d(Y ∩ K, B_2^k) ≤ C(1+ √(k/ln(n/(kln(n+1))))$. This formulation of Dvoretzky's theorem for large dimensional sections is a generalization with a new proof of the result due to Milman and Schechtman for centrally symmetric convex bodies. A sharper estimate holds for the n-dimensional simplex.

2

Random ε-nets and embeddings in $ℓ^{N}_{∞}$

100%

Gordon Y., Litvak A., Pajor A., Tomczak-Jaegermann N.

Studia Mathematica

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2007

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tom 178

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nr 1

91-98

EN

We show that, given an n-dimensional normed space X, a sequence of $N = (8/ε)^{2n}$ independent random vectors $(X_{i})_{i=1}^{N}$, uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $Γ: ℝ → ℝ^{N}$ defined by $Γx = (⟨x,X_{i}⟩)_{i=1}^{N}$ embeds X in $ℓ^{N}_{∞}$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into $ℓ_{∞}^{N}$ with asymptotically best possible relation between N, n, and ε.

3

Banach ideals on Hilbert spaces

62%

Gordon Y., Lewis D. R.

Studia Mathematica

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1975-1976

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tom 54

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nr 2

161-172

Ograniczanie wyników

3 Studia Mathematica

3 Gordon Y.

1 Guédon O.

1 Lewis D. R.

1 Litvak A.

1 Meyer M.

1 Pajor A.

1 Tomczak-Jaegermann N.

1 2007

1 1998

1 1975

An isomorphic Dvoretzky's theorem for convex bodies

Random ε-nets and embeddings in $ℓ^{N}_{∞}$

Banach ideals on Hilbert spaces