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LOG-LIPSCHITZ EMBEDDINGS OF HOMOGENEOUS SETS WITH SHARP LOGARITHMIC EXPONENTS AND SLICING PRODUCTS OF BALLS.
- Source :
-
Proceedings of the American Mathematical Society . Apr2014, Vol. 142 Issue 4, p1275-1288. 14p. - Publication Year :
- 2014
-
Abstract
- If X is a compact subset of a Banach space with X -X homogeneous (equivalently 'doubling' or with finite Assouad dimension), then X can be embedded into some R...n (with n sufficiently large) using a linear map L whose inverse is Lipschitz to within logarithmic corrections. More precisely, there exist c, α > 0 such that c ‖x - y‖/| log ‖x - y‖ |α ≤ |Lx - Ly| ≤ c‖x - y‖ for all x, y ∊ X, ‖x - y‖ < δ, for some δ sufficiently small. It is known that one must have α > 1 in the case of a general Banach space and α > 1/2 in the case of a Hilbert space. It is shown in this paper that these exponents can be achieved. While the argument in a general Banach space is relatively straightforward, the Hilbert space case relies on the fact that the maximum volume of a hyperplane slice of a k-fold product of unit volume N-balls is bounded independent of k (this provides a 'qualitative' generalisation of a result on slices of the unit cube due to Hensley (Proc.AMS 73 (1979), 95-100) and Ball (Proc.AMS 97 (1986), 465-473)). [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 142
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 94330778
- Full Text :
- https://doi.org/10.1090/S0002-9939-2014-11852-1