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Log-Lipschitz embeddings of homogeneous sets with sharp logarithmic exponents and slicing products of balls
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Robinson, James C. (2014) Log-Lipschitz embeddings of homogeneous sets with sharp logarithmic exponents and slicing products of balls. Proceedings of the American Mathematical Society, Volume 142 (Number 4). pp. 1275-1288. doi:10.1090/S0002-9939-2014-11852-1 ISSN 0002-9939.
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Official URL: http://dx.doi.org/10.1090/S0002-9939-2014-11852-1
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 $ \mathbb{R}^n$ (with $ n$ sufficiently large) using a linear map $ L$ whose inverse is Lipschitz to within logarithmic corrections. More precisely, there exist $ c,\alpha >0$ such that
$\displaystyle c\ \frac {\Vert x-y\Vert}{\vert\,\log \Vert x-y\Vert\,\vert^\alpha }\le \vert Lx-Ly\vert\le c\Vert x-y\Vert$$\displaystyle \quad \mbox {for all}\quad x,y\in X,\ \Vert x-y\Vert<\delta ,$
for some $ \delta $ sufficiently small. It is known that one must have $ \alpha >1$ in the case of a general Banach space and $ \alpha >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)).
Item Type: | Journal Article | ||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Journal or Publication Title: | Proceedings of the American Mathematical Society | ||||||||
Publisher: | American Mathematical Society | ||||||||
ISSN: | 0002-9939 | ||||||||
Official Date: | 21 January 2014 | ||||||||
Dates: |
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Volume: | Volume 142 | ||||||||
Number: | Number 4 | ||||||||
Page Range: | pp. 1275-1288 | ||||||||
DOI: | 10.1090/S0002-9939-2014-11852-1 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC) | ||||||||
Grant number: | EP/G007470/1 |
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