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An isomorphic Dvoretzky's theorem for convex bodies

Y. GordonO. GuédonM. Meyer — 1998

Studia Mathematica

We prove that there exist constants C>0 and 0 < λ < 1 so that for all convex bodies K in 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 n satisfying d ( Y K , B 2 k ) C ( 1 + ( k / l n ( n / ( k l n ( 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.

Random ε-nets and embeddings in N

Y. GordonA. E. LitvakA. PajorN. Tomczak-Jaegermann — 2007

Studia Mathematica

We show that, given an n-dimensional normed space X, a sequence of N = ( 8 / ε ) 2 n 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 : N defined by Γ x = ( x , X i ) i = 1 N embeds X in N with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into N with asymptotically best possible relation between N, n, and ε.

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