On the dimension of the \(k\)-medial axis for an arbitrary closed set (Q2089911)

The \(k\)-medial axis \(M_k\) of a closed set \(E \subseteq {\mathbb R}^n\) consists of all points in \({\mathbb R}^n\) that have at least \(k\) nearest points in \(E\) in general position (i.e.\ not all in an affine subspace of dimension \(k-2\)). The author proves for \(1\leq k\leq n+1\) that \(M_k\) is \((n+k-1)\)-rectifiable; in particular, \(M_k\) has Hausdorff dimension at most \(n+k-1\). This is related to a conjecture of \textit{P. Erdős} [Bull. Am. Math. Soc. 51, 728--731 (1945; Zbl 0063.01269)].