Quantitative straightening of distance spheres

TL;DR

This paper demonstrates that, except in dense regions, distance spheres around arbitrary sets in the unit cube can be bi-Lipschitzly straightened into parallel planes, with uniform bounds independent of the set.

Contribution

It introduces a method to decompose the domain into finitely many parts where distance spheres are bi-Lipschitz equivalent to flat planes, regardless of the set's complexity.

Findings

Distance spheres can be straightened into parallel planes outside dense regions.

The decomposition and bi-Lipschitz bounds are independent of the set $K$.

Applicable to arbitrary subsets of the unit cube with controlled density.

Abstract

We study "distance spheres": the set of points lying at constant distance from a fixed arbitrary subset $K$ of $[0,1]^d$. We show that, away from the regions where $K$ is "too dense" and a set of small volume, we can decompose $[0,1]^d$ into a finite number of sets on which the distance spheres can be "straightened" into subsets of parallel $(d-1)$-dimensional planes by a bi-Lipschitz map. Importantly, the number of sets and the bi-Lipschitz constants are independent of the set $K$.