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 ISSN 0951-7715.

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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
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:	Date Event April 2009 Published
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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