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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang


IMPACT FACTOR 2018: 2.316

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1864-8266
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Volume 8, Issue 3

Issues

An example of an infinite Steiner tree connecting an uncountable set

Emanuele Paolini
  • Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
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/ Eugene Stepanov
  • Steklov Mathematical Institute, Russian Academy of Sciences, St. Petersburg Branch, Fontanka 27, 191023 St. Petersburg, Russia; and Department of Mathematical Physics, Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskij pr. 28, Old Peterhof, 198504 St.Petersburg, Russia; and ITMO University, Russia
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/ Yana Teplitskaya
  • Department of Mathematical Physics, Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskij pr. 28, Old Peterhof, 198504 St. Petersburg, Russia
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Published Online: 2015-01-10 | DOI: https://doi.org/10.1515/acv-2013-0025

Abstract

We construct an example of a Steiner tree with an infinite number of branching points connecting an uncountable set of points. Such a tree is proven to be the unique solution to a Steiner problem for the given set of points. As a byproduct we get the whole family of explicitly defined finite Steiner trees, which are unique connected solutions of the Steiner problem for some given finite sets of points, and with growing complexity (i.e. the number of branching points).

Keywords: Steiner problem; infinite tree; explicit solution

MSC: 49Q10; 05C63

About the article

Received: 2013-11-19

Revised: 2014-11-13

Accepted: 2014-12-01

Published Online: 2015-01-10

Published in Print: 2015-07-01


Funding Source: St. Petersburg State University

Award identifier / Grant number: #6.38.670.2013

Funding Source: St. Petersburg State University

Award identifier / Grant number: #6.38.223.2014

Funding Source: Russian Government

Award identifier / Grant number: NSh-1771.2014.1

Funding Source: GNAMPA

Funding Source: RFBR

Award identifier / Grant number: #14-01-00534

Funding Source: Italian Ministry of Research

Award identifier / Grant number: 2010A2TFX2 “Calcolo delle variazioni”


Citation Information: Advances in Calculus of Variations, Volume 8, Issue 3, Pages 267–290, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2013-0025.

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