  If $X$ is a subset of a Banach space with $X-X$ homogeneous, 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,
$$c\,\frac{\|x-y\|}{|\,\log\|x-y\|\,|^\alpha}\le|Lx-Ly|\le c\|x-y\|$$ for all
$x,y\in X$ with $\|x-y\|<\delta$ for some $\delta$ sufficiently small. A simple
argument shows 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 a
result due to Ball (Proc. Amer. Math. Soc. 97 (1986) 465-473) which guarantees
that the maximum volume of hyperplane slices of the unit cube in $\R^d$ is
$\sqrt2$, in dependent of $d$.
