  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 $K$.
