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

TL;DR

This paper demonstrates optimal logarithmic Lipschitz embeddings for homogeneous sets in Banach and Hilbert spaces, achieving sharp exponents and utilizing geometric slicing results.

Contribution

It establishes sharp logarithmic exponents for Lipschitz embeddings of homogeneous sets, extending previous bounds and leveraging geometric slicing techniques.

Findings

Achieves sharp exponent lpha>1 for Banach spaces.

Establishes lpha>1/2 for Hilbert spaces.

Utilizes hyperplane slicing of the unit cube in proofs.

Abstract

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 $d$.