Quantitative straightening of distance spheres

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 $K$.