Quantitative Differentiation and the Medial Axis
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 \(K\).
Fields of Study & Subjects
Advanced Topology and Set Theory
Limits and Structures in Graph Theory
Advanced Harmonic Analysis Research
Publication details
- Journal / source
- Real Analysis Exchange
- Publisher
- Michigan State University Press
- Volume
- 51
- Issue
- 1
- Pages
- 1-10
- Publication date
- 9 April 2026
- Language
- EN