# Existence and regularity results for the Steiner problem

@article{Paolini2013ExistenceAR, title={Existence and regularity results for the Steiner problem}, author={E. Paolini and Eugene O. Stepanov}, journal={Calculus of Variations and Partial Differential Equations}, year={2013}, volume={46}, pages={837-860} }

AbstractGiven a complete metric space X and a compact set $${C\subset X}$$ , the famous Steiner (or minimal connection) problem is that of finding a set S of minimum length (one-dimensional Hausdorff measure $${\mathcal H^1)}$$) among the class of sets
$$\mathcal{S}t(C) \,:=\{S\subset X\colon S \cup C \,{\rm is connected}\}.$$In this paper we provide conditions on existence of minimizers and study topological regularity results for solutions of this problem. We also study the relationships…

## 37 Citations

### On the horseshoe conjecture for maximal distance minimizers

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- 2015

We study the properties of sets $\Sigma$ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\Sigma \subset \mathbb{R}^2$ satisfying the inequality…

### On regularity of maximal distance minimizers

- Mathematics, Computer Science
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This work shows that a maximal distance minimizer is isotopic to a finite Steiner tree even for a "bad" compact $M$, which differs it from a solution of the Steiner problem.

### On uniqueness in Steiner problem

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We prove that the set of $n$-point configurations for which solution of the planar Steiner problem is not unique has Hausdorff dimension is at most $2n-1$. Moreover, we show that the Hausdorff…

### About the optimal pipeline problem

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Consider the class of closed connected sets $\Sigma \subset \mathbb{R}^2$ satisfying the inequality $F_M(\Sigma) \leq r$ for a given compact set $M \subset \mathbb{R}^2$ and some $r > 0$, where
$$…

### A phase-field approximation of the Steiner problem in dimension two

- Mathematics, Computer ScienceAdvances in Calculus of Variations
- 2017

A family of functionals are introduced which approximate the above branched transport energy problem in two dimensions associated with a cost per unit length of the form 1 + β θ {1+\beta\,\theta} , where θ denotes the amount of transported mass.

### Fermat–Steiner problem in the metric space of compact sets endowed with Hausdorff distance

- Materials ScienceJournal of Geometry
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Fermat–Steiner problem consists in finding all points in a metric space Y such that the sum of distances from each of them to the points from some fixed finite subset A of Y is minimal. Such points…

### Approximation of Length Minimization Problems Among Compact Connected Sets

- Mathematics, Computer ScienceSIAM J. Math. Anal.
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This paper provides an approximation of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two, and introduces a term of new type relying on a weighted geodesic distance that forces the minimizers to be connected at the limit.

### Variational Approximation of Functionals Defined on 1-dimensional Connected Sets: The Planar Case

- MathematicsSIAM J. Math. Anal.
- 2018

This paper considers variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem, and provides a variational approximation through Modica-Mortola type energies proving a $\Gamma$-convergence result.

### The Steiner tree problem revisited through rectifiable 𝐺 -currents The Steiner tree problem revisited through rectifiable 𝐺 -currents

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: The Steiner tree problem can be stated in terms of finding a connected set of minimal length containing a given set of finitely many points. We show how to formulate it as a mass-minimization…

### Calibrations for minimal networks in a covering space setting

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In this paper, we define a notion of calibration for an approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover, we introduce the notion of…

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