Wyniki wyszukiwania

1

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

100%

Dmitriyuk A., Gordon Y.

Studia Mathematica

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2009

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tom 195

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nr 3

227-241

EN

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 $m = O(ε^{-2}(k + log p))$ if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map $H: ℝⁿ → ℝ^{m}$ such that for all 1 ≤ l ≤ p and $x,y ∈ W_{l}$ 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.

2

Volume ratios in $L_p$-spaces

100%

Gordon Y., Junge M.

Studia Mathematica

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1999

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tom 136

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nr 2

147-182

EN

There exists an absolute constant $c_0$ such that for any n-dimensional Banach space E there exists a k-dimensional subspace F ⊂ E with k≤ n/2 such that $inf_{ellipsoid ε ⊂ B_E} (vol(B_E)/vol(ε))^{1/n} ≤ c_0 inf_{zonoid Z ⊂ B_F} (vol(B_F)/vol(Z))^{1/k}$ . The concept of volume ratio with respect to $ℓ_p$-spaces is used to prove the following distance estimate for $2≤ q≤ p < ∞$: $sup_{F ⊂ ℓ_p, dim F=n} inf_{G ⊂ L_q, dim G=n} d(F,G) ∼_{c_{pq}} n^{(q/2)(1/q-1/p)}$.

3

Volume approximation of convex bodies by polytopes - a constructive method

81%

Gordon Y., Meyer M., Reisner S.

Studia Mathematica

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1994

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tom 111

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nr 1

81-95

EN

Algorithms are given for constructing a polytope P with n vertices (facets), contained in (or containing) a given convex body K in $ℝ^d$, so that the ratio of the volumes |K∖P|/|K| (or |P∖K|/|K|) is smaller than $f(d)/n^{2/(d-1)}$.

Ograniczanie wyników

3 Studia Mathematica

3 Gordon Y.

1 Dmitriyuk A.

1 Junge M.

1 Meyer M.

1 Reisner S.

1 2009

1 1999

1 1994

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

Volume ratios in $L_p$-spaces

Volume approximation of convex bodies by polytopes - a constructive method