On one-dimensional continua uniformly approximating planar sets

  • Michele MirandaJr.
  • Emanuele Paolini
  • Eugene Stepanov
Article
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Abstract

Consider the class of closed connected sets \(\Sigma\subset {\cal R}^n\) satisfying length constraint \({\cal H}(\Sigma)\leq l\) with given l>0. The paper is concerned with the properties of minimizers of the uniform distance F M of Σ to a given compact set \(M\subset {\cal R}^n\),
$$ F_M(\Sigma):= \max_{y\in M} dist(y,\Sigma), $$
(22)
where dist(y, Σ) stands for the distance between y and Σ. The paper deals with the planar case n=2. In this case it is proven that the minimizers (apart trivial cases) cannot contain closed loops. Further, some mild regularity properties as well as structure of minimizers is studied.

Keywords

Optimal Transportation Minimal Network Energetic Point Homeomorphic Image Optimal Transportation Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Michele MirandaJr.
    • 1
  • Emanuele Paolini
    • 2
  • Eugene Stepanov
    • 3
  1. 1.Dipartimento di Matematica “E. De Giorgi”Università di LecceLecceItaly
  2. 2.Dipartimento di Matematica “U. Dini”Università di FirenzeFirenzeItaly
  3. 3.Dipartimento di Matematica “L. Tonelli”Università di PisaPisaItaly

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