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Log-Lipschitz embeddings of homogeneous sets with sharp logarithmic exponents and slicing products of balls

Author: James C. Robinson
Journal: Proc. Amer. Math. Soc. 142 (2014), 1275-1288
MSC (2010): Primary 37L30, 54H20, 57N60
Published electronically: January 21, 2014
MathSciNet review: 3162249
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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)).

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Additional Information

James C. Robinson
Affiliation: Mathematical Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom

Received by editor(s): September 4, 2011
Received by editor(s) in revised form: May 9, 2012
Published electronically: January 21, 2014
Additional Notes: The author was supported by an EPSRC Leadership Fellowship EP/G007470/1.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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