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}
}
  • E. PaoliniE. Stepanov
  • Published 1 March 2013
  • Mathematics
  • Calculus of Variations and Partial Differential Equations
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… 

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References

SHOWING 1-10 OF 10 REFERENCES

The Steiner Problem for Infinitely Many Points

Let A be a given compact subset of the euclidean space. We consider the problem of finding a compact connected set S of minimal 1dimensional Hausdorff measure, among all compact connected sets

Minimal Networks: The Steiner Problem and Its Generalizations

Some Necessary Results from Graph Theory and Geometry are presented, including the Steiner Problem and Its Modifications.

Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem

Qualitative topological and geometrical properties of optimal networks are studied and a mild regularity result for optimal networks is provided.

Constructing metrics with the Heine-Borel property

A metric space (X, d) is said to be Heine-Borel if any closed and bounded subset of it is compact. We show that any locally compact and ucompact metric space can be made Heine-Borel by a suitable

Hilbert manifolds without epiconjugate points

This theorem had been proved by Myers [6] under the assumption that M was analytic, and this assumption was essential in his proof. Kobayashi, and also Helgason in his book [3], showed that analytic

Steiner's invariants and minimal connections

The aim of this note is to prove that any compact metric space can be made connected at a minimal cost, where the cost is taken to be the one-dimensional Hausdorff measure.

Topics on analysis in metric spaces

1. Some preliminaries in measure theory 2. Hausdorff measures and covering theorems in metric spaces 3. Lipschitz functions in metric spaces 4. Geodesic problem and Gromov-Hausdorff convergence 5.

A Model for the Quasi-Static Growth¶of Brittle Fractures:¶Existence and Approximation Results

Abstract We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G. A. Francfort and J.-J. Marigo, and based on

Topologie, volume 2. Państwowe

  • Wydawnictwo Naukowe, Warszawa,
  • 1958

Russia E-mail address: stepanov.eugene@gmail