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Existence and regularity results for the Steiner problem

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Abstract

Given 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 between several similar variants of the Steiner problem. At last, we provide some applications to locally minimal sets.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Firenze, Firenze, Italy

    Emanuele Paolini

  2. Department of Mathematical Physics, Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskij pr. 28, Old Peterhof, St. Petersburg, Russia, 198504

    Eugene Stepanov

Authors
  1. Emanuele Paolini
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  2. Eugene Stepanov
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Corresponding author

Correspondence to Emanuele Paolini.

Additional information

Communicated by L. Ambrosio.

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Paolini, E., Stepanov, E. Existence and regularity results for the Steiner problem. Calc. Var. 46, 837–860 (2013). https://doi.org/10.1007/s00526-012-0505-4

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  • DOI: https://doi.org/10.1007/s00526-012-0505-4

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