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 $K$.
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