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

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 \[ c\ \frac {\|x-y\|}{| \log \|x-y\| |^\alpha }\le |Lx-Ly|\le c\|x-y\|\quad \mbox {for all}\quad x,y\in X,\ \|x-y\|<\delta ,\] 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)).