Skeletons and central sets

Abstract

Let $\Omega$ be an open proper subset of $\Bbb R^n$. Its {\it skeleton} is the set of points with more than one nearest neighbour in the complement of $\Omega$; its {\it central set} is the set of centres in maximal open balls included in $\Omega$. Intuitively, if we think of $\Omega$ as a land mass in which height is proportional to distance from the sea, its skeleton and central set can be thought of as corresponding to ridges in the mountains of $\Omega$. In this note I discuss the metric and topological properties of such sets. I show that any skeleton in $\Bbb R^n$ is F$_{\sigma}$, and has dimension at most $n-1$, by any of the usual measures of dimension; that if $\Omega$ is bounded and connected, its skeleton and central set are connected; and that $\Omega$ separates $\Bbb R^n$ iff its skeleton does iff its central set does. Any central set in $\Bbb R^n$ is a G$_{\delta}$ set of topological dimension at most $n-1$. In the plane, I show that both skeletons and central sets are locally path-connected, and indeed include many paths of finite length. For any $\Omega$, its central set includes its skeleton; I give examples to show that the central set can be significantly larger than the skeleton.

1991 Mathematics Subject Classification: 54F99.