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Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces
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Robinson, James C. (2009) Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces. Nonlinearity, Vol.22 (No.4). pp. 711-728. doi:10.1088/0951-7715/22/4/001
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Official URL: http://dx.doi.org/10.1088/0951-7715/22/4/001
Abstract
This paper treats the embedding of finite-dimensional subsets of a Banach space B into finite-dimensional Euclidean spaces. When the Hausdorff dimension of X-X is finite, d(H)(X-X) < k, a prevalent set of linear maps from B into R-k are injective on X. The proof motivates the definition of the 'dual thickness exponent', which is the key to proving that a prevalent set of such linear maps have Holder continuous inverse when the box-counting dimension of X is finite and k > 2d(B)(X). A related argument shows that if the Assouad dimension of X-X is finite and k > d(A)(X-X), a prevalent set of such maps are bi-Lipschitz with logarithmic corrections. This provides a new result for compact homogeneous metric spaces via the Kuratowksi embedding of (X, d) into L-infinity(X).
| Item Type: | Journal Article | ||||
|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
| Journal or Publication Title: | Nonlinearity | ||||
| Publisher: | Institute of Physics Publishing Ltd. | ||||
| ISSN: | 0951-7715 | ||||
| Official Date: | April 2009 | ||||
| Dates: |
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| Volume: | Vol.22 | ||||
| Number: | No.4 | ||||
| Number of Pages: | 18 | ||||
| Page Range: | pp. 711-728 | ||||
| DOI: | 10.1088/0951-7715/22/4/001 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access | ||||
| Funder: | Royal Society (Great Britain), Engineering and Physical Sciences Research Council (EPSRC), Royal Society University Research |
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