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If $X$ is a subset of a Banach space with $X-X$ homogeneous, then $X$ can beembedded 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 simpleargument shows that one must have $\alpha>1$ in the case of a general Banachspace and $\alpha>1/2$ in the case of a Hilbert space. It is shown in thispaper that these exponents can be achieved. While the argument in a generalBanach space is relatively straightforward, the Hilbert space case relies on aresult due to Ball (Proc. Amer. Math. Soc. 97 (1986) 465-473) which guaranteesthat the maximum volume of hyperplane slices of the unit cube in $\R^d$ is$\sqrt2$, in dependent of $d$.

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