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 \(M\subset {\cal R}^n\),
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.
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Miranda, M., Paolini, E. & Stepanov, E. On one-dimensional continua uniformly approximating planar sets. Calc. Var. 27, 287–309 (2006). https://doi.org/10.1007/s00526-005-0330-0
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DOI: https://doi.org/10.1007/s00526-005-0330-0