Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T16:46:15.086Z Has data issue: false hasContentIssue false

Skeletons and central sets

Published online by Cambridge University Press:  01 May 1997

Get access

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.

Type
Research Article
Copyright
© London Mathematical Society 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)