Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces. (English) Zbl 1163.54008

Robinson, James C.

Nonlinearity 22, No. 4, 711-728 (2009).

A Borel subset \(S\) of a normed linear space \(V\) is called prevalent if there exists a compactly supported probability measure \(\mu\) such that \(\mu(v + S) = 1\) for all \(v\in V\). This definition was introduced in a slightly different form in [B. R. Hunt, T. D. Sauer and J. A. Yorke, Bull. Am. Math. Soc. (N.S.) 27, No. 2, 217–238 (1992; Zbl 0763.28009)]. For separable spaces an equivalent definition was introduced earlier in [J. P. R. Christensen, Isr. J. Math. 13(1972), Proc. internat. Sympos. partial diff. Equ. Geometry normed lin. Spaces II, 255-260 (1973; Zbl 0249.43002)].
The main purpose of the paper is to study the prevalence of sets of linear maps with finite-dimensional ranges which are one-to-one (or even better) on a given compact subset \(X\) of a Banach space under the assumption that \(X\) is finite-dimensional in a certain metric sense.
Some of the main results:
(1) (Theorem 3.1) Let \(X\) be a compact subset of a Banach space \(B\) such that the Hausdorff dimension of \(X-X\) is \(< k\), where \(k\) is a positive integer. Then a prevalent set of linear maps \(L : B\to\mathbb{R}^k\) are one-to-one between \(X\) and its image.
(2) (Theorem 5.1) For the upper box-counting dimension \(d_B(X)\) the author proves that a prevalent set of such linear maps have Hölder continuous inverses on the image of \(X\) if the box-counting dimension of \(X\) is finite and \(k > 2d_B(X)\).
(3) (Theorem 6.4) For the Assouad dimension \(d_A\) the author of proves that if \(k> d_A(X -X)\), a prevalent set of such linear maps have inverses of the image of \(X\) which are Lipschitz up to a logarithmic term.

Reviewer: Mikhail Ostrovskii (Queens)

MSC:

54C25	Embedding
37C45	Dimension theory of smooth dynamical systems
46B20	Geometry and structure of normed linear spaces
54F45	Dimension theory in general topology

Keywords:

Assouad dimension; box-counting dimension; Hausdorff dimension; prevalent set

Citations:

Zbl 0763.28009; Zbl 0249.43002