Preprint

Inserted: 29 aug 2026
Last Updated: 29 aug 2026

Year: 2026

Abstract:

We provide an example of a planar bounded open set $A$ with countably many connected components that can be connected by adding a set of arbitrarily small positive length (i.e. one-dimensional Hausdorff measure), but cannot be connected by adding any set of zero length. In particular this shows that the existence of a solution to the Steiner problem of finding a set of minimum length connecting a given set cannot be guaranteed when the latter is not compact.

Download:

• counterexampleST.pdf