On the dimension of k-medial axis for arbitrary closed set

TL;DR

This paper proves that the k-medial axis of any closed set in Rn has a rectifiable structure with dimension at most n-k+1, providing a new stratification and answering a longstanding question in the field.

Contribution

It establishes the rectifiability and dimension bounds of the k-medial axis for arbitrary closed sets, offering a novel stratification framework.

Findings

k-medial axis is n-k+1-rectifiable

Provides the first stratification for medial axes of closed sets

Answers a question posed by Erdos

Abstract

We prove that the k-medial axis of an arbitrary closed set in Rn is n-k+1-rectifiable (and hence of dimension at most n-k+1). This result gives a first stratification for medial axis of any closed set, which has been widely studied and used in pure and applied mathematics. This also answers a question proposed by Erdos[4], and leads to more further interesting investigations (see the end of the article).