Open problems on Steiner trees and maximal distance minimizers

TL;DR

This paper reviews open problems in one-dimensional geometric optimization, focusing on maximal distance minimizers and Steiner trees, summarizing known results and highlighting challenging open questions.

Contribution

It compiles and discusses open research questions in geometric optimization problems related to Steiner trees and maximal distance minimizers.

Findings

Some open questions are approachable with elementary methods.

Other questions remain highly challenging and unsolved.

The paper summarizes known results and identifies key open problems.

Abstract

In this work, I collect and discuss a series of open questions in one-dimensional geometric optimization in Euclidean spaces. The focus is on two classes of problems: maximal distance minimizers and Steiner trees. Maximal distance minimizers concern finding a connected set of minimal length whose closed $r$-neighborhood covers a given compact set, whereas Steiner trees aim to find a minimal-length set connecting a prescribed set of points. For both problems, I briefly summarize known results and highlight the remaining open questions. While some questions can be approached with elementary methods, others remain highly challenging.