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

Abstract

If $X$ is a subset of a Banach space with $X-X$ homogeneous, 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, $$c\frac{\Vert x-y\Vert }{|\log \Vert x-y\Vert {|}^{\alpha }}\le |Lx-Ly|\le c\Vert x-y\Vert$$ for all $x,y\in X$ with $\Vert x-y\Vert <\delta$ 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 ${\mathbb{R}}^d$ is $\sqrt{2}$, in dependent of $d$.