Log-Lipschitz embeddings of homogeneous sets with sharp logarithmic exponents and slicing the unit cube

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 $d$.