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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. ISSN 0951-7715

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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

Divisions:

Faculty of Science > Mathematics

Journal or Publication Title:

Nonlinearity

Publisher:

Institute of Physics Publishing Ltd.

ISSN:

0951-7715