Quantitative differentiation and the medial axis

TL;DR

This paper investigates the medial axis of a set in Euclidean space from a quantitative perspective, showing that at most locations and scales, the set appears to fall outside the medial axis when viewed with finite resolution.

Contribution

It introduces a coarse, quantitative analysis of the medial axis, demonstrating that most points and scales in the complement of the set are not close to the medial axis, with bounds independent of the set.

Findings

Most balls in the complement have almost-closest points forming a small angle.

The results are independent of the specific set $K$.

Bounds involve a Carleson packing condition.

Abstract

We study the medial axis of a set $K$ in Euclidean space (the set of points in space with more than one closest point in $K$) from a "coarse" and "quantitative" perspective. We show that on "most" balls $B(x,r)$ in the complement of $K$, the set of almost-closest points to $x$ in $K$ takes up a small angle as seen from $x$. In other words, most locations and scales in the complement of $K$ "appear" to fall outside the medial axis if one looks with only a certain finite resolution. The word "most" involves a Carleson packing condition, and our bounds are independent of the set $K$.