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Open problems on Steiner trees and maximal distance minimizers

Yana Teplitskaya

TL;DR

This survey consolidates open problems in one-dimensional Euclidean optimization, focusing on maximal distance minimizers (MDMs) and Steiner trees. It surveys definitions, known results, and a broad set of open questions across regularity, uniqueness, algebraic structure, density, and high-dimensional nonplanar cases, highlighting connections between the two problem classes and potential asymptotic regimes. The work emphasizes that some questions admit elementary methods while others remain highly challenging, outlining concrete directional questions and conjectures (e.g., horseshoe structure for MDMs, Gilbert–Pollak Steiner ratio, random Steiner tree asymptotics). Taken together, the collection clarifies the boundary between tractable and intractable instances, and underlines the importance of cross-pollination between geometric optimization, algebraic methods, and probabilistic models.

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.

Open problems on Steiner trees and maximal distance minimizers

TL;DR

This survey consolidates open problems in one-dimensional Euclidean optimization, focusing on maximal distance minimizers (MDMs) and Steiner trees. It surveys definitions, known results, and a broad set of open questions across regularity, uniqueness, algebraic structure, density, and high-dimensional nonplanar cases, highlighting connections between the two problem classes and potential asymptotic regimes. The work emphasizes that some questions admit elementary methods while others remain highly challenging, outlining concrete directional questions and conjectures (e.g., horseshoe structure for MDMs, Gilbert–Pollak Steiner ratio, random Steiner tree asymptotics). Taken together, the collection clarifies the boundary between tractable and intractable instances, and underlines the importance of cross-pollination between geometric optimization, algebraic methods, and probabilistic models.

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 -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.

Paper Structure

This paper contains 12 sections, 6 equations, 5 figures.

Table of Contents

  1. Maximal distance minimizers
  2. Explicit examples
  3. Regularity and uniqueness
  4. Algebraic point of view
  5. Steiner trees
  6. Uniqueness and existence
  7. Gilbert--Pollak conjecture on the Steiner ratio
  8. Density type questions
  9. Connection with maximal distance minimizers
  10. Large number of terminals and asymptotical behavior of Steiner trees
  11. Nonplanar explicit examples
  12. Acknowledgements.

Figures (5)

  • Figure 1: A horseshoe for closed curve with big radius of curvature
  • Figure 2: A competitor shorter than the horseshoe for a stadium
  • Figure 3: Example of $6$ points at the half-radius circle
  • Figure 4: A non-unique Steiner tree which is not a Maximal distance minimizer
  • Figure 5: Planar universal Steiner tree